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Edward  3».ri^t- 


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in  2008  with  funding  from 

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ELEMENTS 


DIFFERENTIAL   AND   INTEGRAL 

CALCULUS 

WITH   APPLICATIONS 


BY 

WILLIAM    S.    HALL,  E.M.,  C.E.,  M.S. 

PROFESSOR  OF  TECHNICAL   MATHEMATICS  IN  LAFAYETTE   COLLEGE 

THIRD   EDITION  REVISED 


NEW  YORK 
D.    VAN   NOSTRAND   COMPANY 

1899 


^ 


o3 


u^3 


Copyright,  1897,  ^ 

By  D.  van  NOSTRAND  COMPANY- 


NorfDooD  ^ress : 

J.  S.  Gushing  &  Co.  —  Berwick  &  Smith. 

Norwood,  Mass.,  U.S.A. 


PREFACE. 


This  work  is  an  introduction  to  the  study  of  the  Differential  and 
Integral  Calculus,  and  is  intended  for  colleges  and  technical  schools. 
The  object  has  been  to  present  the  Calculus  and  some  of  its  important 
applications  simply  and  concisely,  and  yet  to  give  as  much  as  it  is 
necessary  to  know  in  order  to  enter  upon  the  study  of  those  subjects 
which  presume  a  knowledge  of  the  Calculus.  The  book  will  be 
found  to  be  adapted  to  the  needs  of  the  mathematical  student,  and 
also  will  enable  the  engineer  to  get  that  knowledge  of  the  Calculus 
which  is  required  by  him  in  order  to  make  practical  applications  of 
the  subject. 

All  of  the  formulas  for  differentiation  are  established  by  the 
method  of  limits.  This  method  is  preferred  because  it  is  more  readily 
understood,  and  is  more  rigorous  than  the  method  of  infinitesimals; 
and,  moreover,  it  has  the  great  advantage  of  being  a  familiar  method, 
as  the  student  has  previously  used  it  in  Algebra  and  Geometry. 
But  the  differential  notation  is  fully  explained,  and  is  employed 
when  there  is  any  advantage  gained  by  so  doing,  particularly  in  the 
investigations  of  the  Integral  Calculus. 

As  soon  as  the  fundamental  formulas  of  differentiation  have  been 
established,  the  corresponding  inverse  operations  or  integrations  fol- 
low. Thus  the  essential  unity  of  the  two  branches  of  the  Calculus 
is  emphasized,  the  whole  subject  is  made  more  intelligible,  and  there 
is  a  saving  of  much  space. 

Principal  applications  of  the  Calculus,  as  in  Maxima  and  Minima, 


797946 


iv  PREFACE. 

Radius  of  Curvature,  etc.,  are  treated  at  some  length,  while  less 
important  subjects  are  treated  much  more  briefly. 

A  large  number  of  carefully  selected  examples,  some  original  ones, 
and  numerous  practical  numerical  problems  from  mechanics  and  dif- 
ferent branches  of  applied  mathematics  are  given. 

As  there  has  been  an  increasing  demand  for  a  short  course  in 
Differential  Equations,  a  chapter  on  this  subject  is  given  which  it  is 
hoped  will  meet  a  much-felt  want. 

A  table  of  Integrals,  arranged  for  convenience  of  reference,  is 
appended. 

Many  American  and  English  books,  and  some  of  the  leading 
French  and  German  works,  have  been  freely  consulted,  and  problems 
have  been  gathered  from  many  different  sources. 

WILLIAM  S.  HALL. 
Easton,  Pa.,  January,  1897. 


CONTENTS 


CHAPTER  I. 

Definitions  and  First  Pbinciples. 

Art.  Page 

1.  Constants  and  Variables 1 

2.  Functions _1 

3.  Increments 3 

4.  Limits        . 4 

6.     Theory  of  Limits • .        .        .        .        .  5 

6.  Limiting  Ratio  of  Increments 6 

7.  Derivatives .  8 

8.  Differentiation  and  the  Differential  Calculus 10 


CHAPTER  II. 

Differentiation  of  Algebraic  Functions. 

9.     Definition 11 

10.  Algebraic  Sum  of  Functions 11 

11.  Product  of  a  Constant  and  a  Function 12 

12.  Any  Constant 12 

13.  Product  of  Two  Functions         .        . 13 

14.  Product  of  Three  or  More  Functions 13 

15.  Quotient  of  Two  Functions ,        .  14 

16.  Constant  Power  of  a  Function.     Problems 15 

CHAPTER   in. 
Differentiation  of  Transcendental  Functions. 

17.  Definitions .  19 

18.  Base  of  the  Natural  System  of  Logarithms 20 

19.  Logarithmic  Functions 20 

V 


Vi  CONTENTS. 

Art.  Page 


20.  Exponential  Function  with  Constant  Base 

21.  Exponential  Function  with  Variable  Base.     Problems 

22.  Circular  Measure 

23.  Limiting  Value  of  ^^. 


24.     Trigonometric  Functions 

26.     Inverse  Trigonometric  Functions.    Problems  . 


21 
22 
24 

25 

26 
29 


CHAPTER   IV. 
Differentials. 

26.  Definition o        .        .        .      34 

27.  Geometric  Interpretation  of  -^ ,        .        .35 

dx 

28.  Geometric  Derivation  of  Formulas.    Problems         .        .        c        .        .      36 


CHAPTER  V. 
Integration. 

29.  Definition 40 

30.  Fundamental  Formulas 40 

31.  Elementary  Rules 42 

32.  Constant  of  Integration.     Problems 43 

33.  Integration  of  Trigonometric  Differentials.     Problems     ....  48 

34.  Definite  Integrals •        .  49 

36.     Geometric  Illustration  of  Definite  Integration 60 

36.     Change  of  Limits.     Problems    .        . 62 


CHAPTER   VL 
Successive  Differentiation  and  Integration. 

37.  Successive  Derivatives 55 

38.  Successive  Integration.     Problems    .        . 66 

Applications  in  Mechanics. 

39.  Velocity  and  Acceleration  of  Motion         .  59 

40.  Uniformly  Accelerated  Motion.     Problems  60 

41.  Derivatives  of  the  Product  of  Two  Functions.    Problems         .        .        .61 


CONTENTS.  vii 


CHAPTER   VII. 

FUNCTIONB   OF   TwO   OR    MORE    VARIABLES.       IMPLICIT   FUNCTIONS.      ChANGE  OF 

THE  Independent  Variable. 

A  KT.  Page 

42.  Partial  Differentiation 64 

43.  Total  Differential  of  a  Function  of  Two  or  More  Independent  Variables .  65 

44.  Total  Derivative  when  M  =/(x,  y,  2),  ?/  =  0(x),  and  2  =  01  (a;)         .         .  66 

45.  Successive  Partial  Derivatives 68 

46.  If  ?t=/(a;,  y),toprovetliat-^=  ^^  ......  69 

dydx    dxdy 

47.  Implicit  Functions,    Problems 69 

48.  Integration  of  Functions  of  Two  or  More  Variables         ....  71 

49.  Integration  of  Total  Differentials  of  the  First  Order.    Problems       .         .  72 

50.  Change  of  the  Independent  Variable.     Problems 73 


CHAPTER   VIII. 
Development  of  Functions. 

51.  Definition 78 

52.  Maclaurin's  Tneorem.    Problems      .         .        .        .        .        .  .78 

53.  Taylor's  Theorem ,         .  83 

54.  Demonstration  of  lay lor's  Theorem.    Problem 84 

55.  Rigorous  Proof  of-  Taylor's  Theorem 87 

56.  Remainder  in  Taylor's  and  Maclaurin's  Theorems 88 

57.  Taylor's  Theorem  for  Functions  of  Two  or  More  Independent  Variables  89 

CHAPTER   IX. 
Evaluation  of  Indeterminate  Forms. 

58.  Indeterminate  Forms 91 

59.  Functions  that  take  the  Form  -.     Problems     -. 92 

0 

60.  Functions  that  take  the  Form  - 94 

CO 

61.  Functions  that  take  the  Forms  0  x  qo  and  oo  —  oo 95 

62.  Functions  that  take  the  Forms  0°,  00°  and  li"'.     Problems       ...  96 

63.  Compound  Indeterminate  Forms.    Problems   ......  98 


CHAPTER   X. 
Maxima  and  Minima  of  Functions. 

64.  Definitions  and  Geometric  Illustration      .        .        ,        ,        .        .        .      99 

65.  Method  of  Determining  Maxima  and  Minima  .        .  .     ,        .        .        .     100 


VlU  CONTENTS. 

Art.  Page 

66.  Conditions  for  Maxima  and  Minima  by  Taylor's  Theorem.    Problems     .  101 

67.  Maxima  and  Minima  of  Functions  of  Two  In  1  pendent  Variables   .         .  Ill 
Q8.  Maxima  and  Minima  of  Functions  of  Tbiv  e  Independent  Variables. 

Problems >     113 


CHAPTER   XI. 

Tangents,  Normals  and  Asymptotes. 

69.  Equations  of  the  Tangent  and  Normal 116 

70.  Lengths  of  Tangent,  Normal,  Subtangent  and  Subnormal         .         .        .  117 

71.  Tangent  of  the  Angle  between  the  Radius  Vector  and  the  Tangent  to  a 

Plane  Curve  in  Polar  Coordinates 117 

72.  Derivative  of  an  Arc 118 

73.  Derivative  of  an  Arc  in  Polar  Coordinates 119 

74.  Lengths  of  Tangent,  Normal,  etc.,  in  Polar  Coordinates.     Problems        .  120 

75.  Rectilinear  Coordinates 122 

76.  Asymptotes  parallel  to  the  Axis        ........  123 

77.  Asymptotes  determined  by  Expansion 124 

78.  Asymptotes  in  Polar  Coordinates.    Problems 124 


CHAPTER   XII. 

Direction  of  Curvature.    Points  of  Inflection.     Radius  of  Curvature. 

Contact. 

79.  Direction  of  Curvature 126 

80.  Direction  of  Curvature  in  Polar  Coordinates 128 

81.  Points  of  Inflection.     Problems 129 

82.  Curvature.    Problems 130 

83.  Radius  of  Curvature 131 

84.  Radius  of  Curvature  in  Polar  Coordinates 132 

85.  Contact  of  Different  Orders      .         . 133 

86.  Radius  of  Osculating  Circle  and  Coordinates  of  Centre     .        .        .        .134 

87.  Osculating  Circle  and  Contact  of  the  Third  Order,     Problems          .        .  136 


CHAPTER  XIIL 

EVOLUTES   AND   INVOLUTES.      ENVELOPES. 

88.  Definition 138 

89.  Equation  of  the  Evolute 138 

90.  A  Normal  to  Any  Involute  is  Tangent  to  its  Evolute        .        .        .        .140 


CONTENTS.  ix 

Art.  Page 

91.  The  Difference  between  any  Two  Radii  of  Curvature  of  an  Involute      .     141 

92.  Mechanical  Construction  of  an  Involute 142 

93.  Envelopes  of  Curves 142 

94.  Equation  of  the  Envelope  of  a  Family  of  Curves.    Problems  .        .     143 


CHAPTER   XIV. 
Singular  Points. 

95.  Definitions 147 

96.  Multiple  Points.    Problems 147 

97.  Cusps 150 

98.  Conjugate  Points.    Stop  Points.    Shooting  Points 151 

CHAPTER   XV. 

Integration  of  Rational  Fractions. 

99-    Rational  Fractions.    Problems .154 

CHAPTER   XVI. 

Integration  of  Irrational  Functions. 

100.  Irrational  Functions 160 

101.  Irrational  Functions  containing  only  Monomial  Surds.    Problems  .     160 

102.  Functions  containing  only  Binomial  Surds  of  the  First  Degree.    Prob- 

lems     161 

103.  Functions  having  the  Form     ^^"^^^^^ 162 

(a  +  6a;2)2 

104.  Functions  having  the  Form /(x,^^^_±_^  j  da;:    Problems      .        .        .     163 

^  CX  ~\~  CI  ^ 

105.  Functions  containing  only  Trinomial  Surds  of  the  Form  Va  -hbx  +  cx^. 

Problems 164 

106.  Binomial  Differentials 166 

107.  Conditions  of  Integrability  of  Binomial  Differentials.    Problems   .  166 

CHAPTER   XVII. 
Integration  by  Parts  and  by  Successive  Reduction. 

108.  Integration  by  Parts.     Problems .     170 

109.  Formulas  of  Reduction.     Problems  .         .         .         .         .         .         ,     171 


X  CONTENTS. 

CHAPTER  XVin, 

Integration  of  Transcendental  Functions.    Integration  by  Series. 

Aet.  Page 

110.     Introduction 177 

177 

178 

179 

182 
182 


111.  ^Integration  of  the  Form  (/(a:)(logx)"da:.     Problems     . 

112.  Integration  of  the  Form  i  x'^a'^dx.     Problems 

113.  Integration  of  the  Form  j  sin"»5  cos"i?  fZ^.     Problems 

114.  Integration  of  the  Forms  i  x«  Q,o&{ax)dx  and  (  x"  s,m{ax)dx 

115.  Integration  of  the  Forms  i  f«^sin»a:d:«  and  j  e»*  cos"  a;  (?x.     Problems   , 

116.  Integration  of  the  Forms  \f{x)  arc  sin  x  dx,  \f{x)  arc  cos  x  dx,  etc 

Problems 

Jdd 
Problems  .... 
a  +  6  cos  ^ 

118.     Integration  by  Series.     Problems 


184 
184 
186 


CHAPTER  XIX. 
Integration  as  a  Summation.     Areas  and  Lengths  of  Plane  Curves. 

119.  Integration  as  a  Summation.     Problems 187 

120.  Areas  of  Plane  Curves  in  Polar  Coordinates.     Problems  .         .         ,  191 

121.  Rectification  of  Plane  Curves  referred  to  Rectangular  Axes.     Problems  192 

122.  Rectification  of  Curves  in  Polar  Coordinates.     Problems         .         .        .  196 

123.  The  Common  Catenary 196 

CHAPTER  XX. 
Surfaces  and  Volumes  of  Solids. 

124.  Surfaces  and  Volumes  of  Solids  of  Revolution      Problems     .         .        .     199 

125.  Surfaces  by  Double  Integration 202 

126.  Volumes  by  Triple  Integration.     Problems      .        .        ....    203 

CHAPTER  XXI. 

I 
Centre  of  Mass.     Moment  of  Inertia.     Properties  of  Guldin. 

127.  Definitions 206 

128.  General  Formulas  for  Centre  of  Mass.     Problems  .        .        .        .        .     207 

129.  Centre  of  Mass  for  Plane  Surfaces.     Problems 210 


COXTENTS.  xi 

Art.  Page 

130.  Centre  of  Mass  of  Surfaces  of  Revolution.     Problems     ....     212 

131.  Centre  of  Mass  of  Solids  of  Revolution.     Problems         .         .         .         .213 

132.  Moments  of  Inertia  of  Surfaces.     Problems 214 

133.  Guldin's  Theorems.    Problems 215 


CHAPTER  XXII. 

Differential  Equations. 


134.  Definition 

135.  Differential  Equations  of  the  First  Order  and  Degree.    Problems  . 

136.  Homogeneous  Differential  Equations.     Problems    .... 

137.  The  Form  {ax  -\- hy  -\-  c)dx  +  (a'x  +  biy  +  c')dy  =  0.     Problems    . 

138.  The  Linear  Equation  of  the  First  Order.     Problems 

139.  Extension  of  the  Linear  Equation.     Problems         .... 

140.  Exact  Differential  Equations.     Problems 

141.  Factors  Necessary  to  make  Differential  Equations  Exact.     Problems 

142.  First  Order  and  Degree  with  Three  Variables.     Problems 

143.  First  Order  and  Second  Degree.     Problems 

144.  Differential  Equations  of  the  Second  Order.     Problems 


217 
218 
219 
221 
222 
224 
225 
227 
230 
232 
234 


APPENDIX. 

Table  of  Integrals 238 

Note  A 250 


DIFFEEENTIAL  AND  INTEGEAL  CALCULUS. 

CHAPTER   I. 
DEFINITIONS  AND  FIRST  PRINCIPLES. 
Art.  1.     Constants  and  Variables. 

The  quantities  employed  in  the  Calculus  belong  to  two  classes, — 
constants  and  variables. 

A  constant  quantity  is  one  which  retains  the  same  value  through- 
out the  same  discussion.  Constants  are  usually  denoted  by  the  first 
letters  of  the  alphabet. 

A  variable  quantity  is  one  which  admits  of  an  infinite  number 
of  values  in  the  same  discussion  within  limits  determined  by  the 
nature  of  the  problem.  Variables  are  usually  represented  by  the  last 
letters   of  the   alphabet. 

Art.  2.     Functions. 

One  variable  quantity  is  a  function  of  another  when  they  are  so 
related  that  for  any  assigned  value  of  the  latter  there  is  a  corre- 
sponding value  of  the  former.  Arbitrary  values  may  be  assigned  to 
the  second  variable,  which  is  then  called  the  independent  variable, 
while  the  first  variable  or  function  is  called  the  dependent  variable. 

For  example,  the  area  of  a  circle  is  a  function  of  its  diameter 
because  the  area  depends  on  the  length  of  the  diameter,  and  the 
diameter  whose  length  may  be  assigned  at  pleasure  is  the  inde- 
pendent variable. 

B  1 


2    .    ,  ...  J>IEFERENTIAL  "AND  INTEGRAL  CALCULUS. 

The  trigonometric  functions  are  functions  of  the  angle,  the  angle 
being  regarded  as  the  independent  variable. 

^    Expressions  involving  a;,  such  as 


oc^,  ax^  -\-  bx  +  c,  log  X,  Vl  —  a?*, 

•- 

are  functions  of  the  independent  variable  x. 

a 

A  quantity  may  be  a  function  of  two  or  more  variables.  For  exam- 
ple, the  area  of  a  plane  triangle  is  a  function  of  its  base  and  altitude ; 
the  volume  of  a  rectangular  parallelopiped  is  a  function  of  its  three 
dimensions. 

The  expressions, 


a^-Zx'if^y^    ^o'x^  +  hY,   a^'+y, 
are  functions  of  x  and  y. 
The  expressions, 

a'^r^j^lY^(?z%   lo^ix'^-xy  +  z^ 

are  functions  of  x,  y,  and  z. 

An  explicit  function  is  one  whose  value  is  expressed  directly  in 
terms  of  the  independent  variable  and  constants.  For  example,  y  is 
an  explicit  function  of  x  in  the  equations 


2/  =  - Va^  —  d\  and  y  =  2ax-\-:»?  +  a?. 
Explicit  functions  are  denoted  by  such  symbols  as  the  following : 

which  may  be  read  respectively:  "?/  equals  the  /  function  of  a;";  "2/ 
equals  the  large  F  function  of  x" \  "1/  equals  the  <^  function  of  a; " ; 
"?/  equals  the /prime  function  of  x." 

When  the  equation  giving  the  relation  connecting  the  variables  is 
not  solved  with  reference  to  y,  y  is  an  implicit  function  of  x.  For 
example,  y  is  an  implicit  function  of  x  in  the  equations 


DEFINITIONS  AND  FIRST  PRINCIPLES.  3 

Implicit  functions  are  denoted  by  such  symbols  as  the  following : 

fix,y)  =  0;  F(x,y)  =  0',    ^{x,y)  =  0', 
which  may  be  read,  "  the  /  function  of  x  and  y  equals  zero  " ;  etc. 

Art.  3.     Increments. 

If  a  variable  receives  any  addition  to  its  value,  this  addition  is 
called  an  increment,  and  is  usually  denoted  by  the  symbol  A  placed 
before  the  variable.  Thus  an  increment  received  by  the  variable  x 
would  be  denoted  by  ^x,  and  would  be  read  "  delta  a;,"  or  "  increment 
of  ic."  The  increment  of  a  variable  may  be  either  positive  or  negative ; 
if  it  is  positive  the  variable  is  increasing,  and  if  it  is  negative  the 
variable  is  decreasing.  A  negative  increment  is  sometimes  called  a 
decrement. 

Art.  4.     Limits. 

A  limit  of  a  variable  is  a  constant  value  which  the  variable  contin- 
ually approaches,  and  from  which  it  can  be  made  to  differ  by  a  quan- 
tity less  than  any  assignable  quantity,  but  which  it  cannot  absolutely 
equal. 

For  example,  assume  that  a  body  is  moving  along  a  straight  line 
from  ^  to  5  as  in  Fig.  1,  under  the  condition  that  in  the  first  interval 

[  '  i  2  3        4       i 

Fig.  1. 

of  time  it  shall  move  one-half  of  the  entire  distance,  or  from  A  to  1, 
and  one-half  of  the  remaining  distance,  or  from  1  to  2,  in  the  second 
interval,  and  so  on,  moving  during  each  interval  one-half  of  the  dis- 
tance remaining.  In  this  case  the  entire  distance  AB  is  a  constant 
toward  which  the  distance  traversed  by  the  moving  point  continually 
approaches  as  a  limit  but  never  reaches. 

The  limit  of  -,  as  ic  increases  indefinitely,  is  zero ;  as  a;  in  this  frac- 

tion  increases  the  fraction  decreases,  and  as  x  may  be  increased  at 
pleasure,  the  fraction  may  be  made  to  approach  indefinitely  near  to 
zero. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Let  the  locus  of  the  equation  ^  =  -  be  drawn  by  the  method  of 


rectangular  coordinates  as  in  Fig.  2. 


Fig.  2. 


If  x=  1,  then  y  =  1; 

If  x  =  2,  then  y  =  .5-, 

If  x  =  4t,  then  y  =  .25 ; 

If  a;  =  100,  then  ?/  =  .01 ; 


Or  as  the  abscissa  increases  the  ordinate  decreases  toward  zero  as 
a  limit ;  thus  the  curve  continually  approaches  the  X-axis,  but  never 
reaches  it. 

The  limit  of  the  value  of  the  repeating  decimal  0.555...,  as  the 
number  of  decimal  places  is  continually  increased,  is  f. 

A  variable  may  approach  its  limit  in  three  ways : 

1st.  A  variable  may  increase  toward  its  limit,  as  is  the  case  when 
a  polygon  is  inscribed  in  a  circle ;  the  polygon  will  increase  toward  the 
circle  as  its  limit,  as  the  number  of  sides  is  increased. 

2d.  A  variable  may  decrease  toward  its  limit,  as  is  the  case  when 
a  polygon  is  circumscribed  about  a  circle;  the  polygon  will  decrease 
toward  the  circle  as  its  limit,  as  the  number  of  sides  is  increased. 

3d.  A  variable  may  approach  its  limit  and  be  sometimes  greater 
and  sometimes  less  than  its  limit.     For  example,  take  the  geometrical 


DEFINITIONS  AND  FIRST   PRINCIPLES.  5 

progression  whose  first  term  is  1  and  whose  ratio  is  —  ^,  giving  the 
series  1,  —  J,  J?  —  tt>  •  •  • ;  ^^^^  ^^^  limit  of  the  sum  of  the  series,  as 
the  number  of  terms  is  indefinitely  increased,  is  |;  but  the  sum  of 
any  odd  number  of  terms  will  be  greater  than  this  limit,  and  the  sum 
of  any  even  number  of  terms  will  be  less. 

Art.  5.     Theory  of  Limits. 

From  the  definition  of  a  limit  of  a  variable,  it  follows  that  the 
difference  between  the  variable  and  its  limit  is  a  variable  which  has 
zero  for  its  limit.  Therefore,  to  prove  that  a  given  constant  is  the 
limit  of  a  certain  variable,  it  is  sufficient  to  show  that  the  difference 
between  the  variable  and  the  constant  has  the  limit  zero. 

1st.  The  fundamental  proposition  in  the  theory  of  limits  is  the 
^ollowin<^: 

If  two  variables  are  equal  and  are  so  related  that  as  they  change  they 
remain  always  equal  to  each  other,  and  each  a2:)proaches  a  limit,  their  limits 
are  equal. 

Let  X  and  y  be  the  variables,  and  a  and  h  their  respective  limits, 
and  let  a?'  and  y'  represent  the  differences  between  the  variables  and 
their  limits. 

Then  a  =  x  +  x\   and  b  =  y  -^y'. 

Since  x  =  y  is  always  true, 

a  —b  =  x'  —  y'.  (1) 

In  equation  (1),  x'  —  y'  is  equal  to  a  constant,  and  x'  and  y'  are  varia* 
bles  that  approach  zero  as  a  limit.  Hence  x'  —  y'  =  0,  and,  therefore, 
a  —  b  =  0,  ov  a  =  b. 

The  supplementary  propositions  are  readily  established. 

2d.  The  limit  of  the  algebraic  sum  of  a  finite  number  of  variables 
is  the  algebraic  sum  of  their  limits. 

3d.  The  limit  of  the  product  of  two  or  more  variables  is  the  product 
of  their  limits. 

4th.  The  limit  of  the  quotient  of  two  variables  is  the  quotient  of 
their  limits. 


6  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

Art.  6.     Limiting  Ratio  of  Increments. 

^  If  an  increment  be  given  to  a;  in  2/  =f(x),  y  will  receive  a  corre- 
sponding increment ;  required  the  limiting  value  of  the  ratio  — ^• 

Aa; 
Taking  first  a  particular  function,  for  example,  y  —  ax^. 

In  this  example,  if  x  receives  an  increment  represented  by  ^x  or  /i, 
y  will  take  a  corresponding  increment  represented  by  Ay]  and  the 
equation  becomes 

y-\-Ay  =  a(x-\-hy 

=  ax^  -\-2  axil  -f-  aAl 

Subtracting  y  =  aa^^ 

Ay  =  2axh-\-ah\  (1) 

Dividing  by  Ax  =  h, 

^=2ax-i-ah.  (2) 

Ax 

As  7i  approaches  zero,  each  member  of  this  equation  will  approach 
a  limit,  and  by  Art.  5  these  limits  are  equal ;  therefore 

limit  of  ^  =  2  ax.  (3) 

Ax  ^  ^ 

In  order  to  make  a  definite  application,  let  a  =  16  in  the  given  equa- 
tion, and  substitute  s  for  y,  and  t  for  x\  then  the  equation  becomes 
s  =  16  fy  which  is  approximately  the  equation  of  a  freely  falling  body 
near  the  earth's  surface,  s  representing  the  number  of  feet  fallen,  and  t 
the  time  of  the  fall  in  seconds. 

Then  the  proper  substitutions  made  in  equation  (3)  give 

limit  of —  =  32^, 

which  is  seen  to  be  the  actual  velocity  at  the  end  of  t  seconds.  There- 
fore the  limiting  ratio  of  the  increments  of  distance  and  time  is  the 
velocity  at  the  end  of  the  period. 

To  illustrate  further,  let  the  object  be  to  determine  the  increments 
produced  in  s  by  certain  decreasing  increments  assigned  to  t,  when  t 
has  some  given  value,  as  10. 


DEFINITIONS   AND   FIRST   PRINCIPLES.  7 

Substituting   s  =  2/,    ^  =  a;  =  10,    a  =  16   and   At  =  Aa;,   in    (1),   (2) 
and  (3): 

As  =  320At-\-16(M)% 

|^  =  320  +  16(A0, 

and  limit  of  —  =  320. 

At 

Let  At  =  0.1,        then  As  =  32.16  and  —  =  321.6 ; 

A^  ' 

Let  A^  =  0.01,      then  As  =    3.2016  and  —  =  320.16  : 

A^  ' 

Let  A^  =  0.001,    then  As  =      .320016      and —  =  320.016  ; 

At  ' 

Let  A^  =  0.0001,  then  As  =      .03200016  and  —  =  320.0016  ; 

A^  ' 


And  it  is   apparent  that  as   A^  continually  diminishes,  As  also 

As 

decreases,  and  the  ratio  of  the  increments,  — ,  approaches  320  as  its 

limit. 

Take  next  a  geometrical  example. 

Let  the  curve  be  the  parabola  whose  equation  is  y  =  V2  x^  and 
whose  locus  is  shown  in  Fig.  3.  Let  (x',  y')  be  the  coordinates  of  P, 
and  (x'  +  Ax,  y'  +  Ay)  be  the  coordinates  of  any  second  point  P'. 


Then  y' +  Ay  =  ^2  (x' -\- Ax).  (1) 

Subtracting        q^  y'=^2x', 

^-y^^  Ay  =  ^2(x'  +  Ax)-V2^', 

hence   "  '^^  Ay^V2(x'  +  Ax)-V2^'^  ^ 

Ax  Ax  ^  ^ 

Rationalizing  the  numerator  of  (2), 

Ay  2  Air  2 

— ^  = — — —  =  —  — ; i    :s^ 

Aa;     Aa;[V2(x'  +  Aa;)  + V2a;']      V2  (x' -^  Ax) -{- ^/2x'  ^^ 

and  limitof  ^  =  — 4=  =  -^  =  i.      ^^ 

^a;     2V2a;'      V2x'     y\^' 


8  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

From  the  figure,  it  is  obvious  that  — ^  is  the  tangent  of  the  angle 

Ax 

P^TN,  and  if  the  point  P'  approaches  indefinitely  near  to  P,  the  line 

PT  will  be  a  tangent  to  the  curve  at  P.     Therefore,  the  limit  of  -^,  as 

Ax 


Fig.  3. 


Ax  approaches  zero,  is  the  tangent  of  the  angle  which  the  curve  makes 
with  the  X-axis,  and  is  equal  to  the  reciprocal  of  the  ordinate  of  the 
point  of  contact. 

If  P  is  at  the  extremity  of  the  latus  rectum,  coordinates  (-J,  1),  then 


limit  of  ^ 
Ax 


tan  45= 


or  the  tangent  to  the  parabola  at  the  extremity  of  the  latus  rectum 
makes  an  angle  of  45°  with  the  X-axis,  which  is  a  well-known  property 
of  the  curve. 


Art.  7.     Derivatives. 

The  limit  of  — ^  in  the  preceding  article  is  called  the  derivative  of  y 

^^     .  dv 

with  respect  to  x,  and  is  denoted  by  —  •     Hence  the  definition :  If  y  is 

cix 

a  function  of  x,  the  derivative  of  y  with  respect  to  x  is  the  limiting 


DEFINITIONS  AND  FIRST  PRINCIPLES.  9 

value  of  the  ratio  of  the  increment  of  y  to  the  corresponding  increment 
of  ic,  as  the  increment  of  x  approaches  zero. 

In  general,  let  y=zf{x).  (1) 

When  X  is  given  an  increment  Aa?  or  h,  y  takes  a  corresponding 
increment  Ay,  and  the  equation  becomes 

y  +  ^y=f{x  +  h).  (2) 

Subtracting  (1)  from  (2), 

^y=f{x  +  h)-f{x)._  (3) 

Dividing  (3)  by  Ax,        Ay  ^/(o;  + /^ -/(a;)^  ^^^ 

As  Ax  approaches  zero,  the  limit  of  — ^  is  the  derivative  of  the 

dv  ^^ 

function,  and  is  represented  by  -^« 

dx 

Therefore       ^^  =  SM  =  limit  of  fi^  +  K)~m_ 
dx        dx  Ax 

The  derivative  is  often  called  the  differential  coefficient,  and  the 
symbol /'(x)  is  frequently  used  instead  of  --^• 

The  term  "derivative  "  is  fully  as  significant  as  differential  coefficient,  and  is 
certainly  to  be  preferred  when  the  method  of  limits  is  used.  The  word  "deriva- 
tive "  will  generally  he  used  in  this  book.  Derived  function  is  another  name  which 
is  sometimes  adopted  instead  of  the  word  "derivative."  It  must  be  carefully 
noticed  that  A  and  d  are  not  factors,  but  symbols  of  operations. 

The  general  method  of  finding  the  derivative  of  any  function  of  x  is 
as  follows :  Two  values  of  the  independent  variable,  as  x  and  x  +  Ax, 
are  taken,  and  the  corresponding  values  of  the  given  function  are 
found;  the  difference  between  these  two  values  of  y  is  the  increment 
of  the  function  corresponding  to  the  increment  Ax  given  to  x.  The 
limit  of  the  ratio  of  these  two  increments,  as  Ax  approaches  zero,  will 
be  the  derivative  of  the  function. 

According  to  this  method,  general  rules  or  formulas  are  obtained 
for  forming  the  derivatives  of  the  different  kinds  of  functions. 


10  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Art.  8.    Differentiation  and  the  Differential  Calculus.* 

The  operation  of  finding  the  derivative  of  a  function  is  called 
differentiation. 

The  object  of  the  Differential  Calculus  is  to  determine  the  deriva- 
tives of  functions,  and  to  investigate  their  properties  and  applications. 

f  J  PROBLEMS. 

1.  In  the  equation  2/  =  £c^  —  3ic  +  5,  what  is  the  increment  received 
by  y  if  an  increment  of  1  is  given  to  x,  when  ic  =  3  ?  Ans.  4. 

2.  In  the  equation  y  —  mx  -\-  n,  what  is  the  ratio  of  the  increment 
of  the  ordinate  to  the  increment  of  the  abscissa  ?  A^is.  m. 

3.  In  the  equation  y^  =  ^(ISx  —  x^),  what  are  the  increments 
received  by  y  corresponding  to  an  increment  of  1  given  to  the  abscissa, 
starting  from  x  =  S?  Ans.   —  2 V5  ±  fVli. 

*  The  Differential  and  Integral  Calculus  originated  in  the  seventeenth  century. 
Newton  was  the  first  discoverer  of  the  new  analysis,  but  to  Leibnitz  belongs  the 
credit  of  priority  of  publication  and  the  invention  of  a  notation  much  superior  to 
Newton's,  and  which  has  entirely  superseded  it.  Leibnitz  first  published  his  new 
method  in  1684. 

Newton  called  his  method  the  method  of  fluxions.  According  to  him,  all  quan- 
tities are  supposed  to  be  generated  by  continuous  motion,  as  a  line  by  a  moving 
point.  Fluxions  are  the  relative  rates  with  which  functions  and  the  variables  on 
which  they  depend  are  increasing  at  any  instant. 

Leibnitz  considered  all  quantities  to  be  made  up  of  indefinitely  small  parts  or 
infinitesimals ;  a  surface  being  composed  of  indefinitely  small  parallelograms,  and 
a  volume  of  indefinitely  small  parallel opipeds.  The  nomenclature  and  notation  of 
the  Calculus  now  in  common  use  were  original  with  Leibnitz,  and  introduced  by 
him. 

According  to  Newton,  the  fluxion  of  x  would  be  denoted  by  x,  while  by  Leibnitz 

dx 
the  corresponding  derivative  is  — 

dt 


> 


-.u 


11  ^ 


f^  ;.%  ^^ 


4^1" 


CHAPTER   II. 

DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS. 
Art.  9.     Defixition. 

An  algebraic  function  is  one  in  which  the  only  indicated  operations 
are  addition,  subtraction,  multiplication,  division,  and  involution  and 
evolution  with  constant  exponents.* 

In  this  chapter  only  functions  of  a  single  independent  variable  x 
will  be  treated,  and  throughout  the  chapter  u,  v  and  w  will  be  regarded 
as  functions  of  x.  .^ 

Art.  10.     Algebraic  Sum  of  Any  Number  of  Functions. 

If  y  be  taken  to  represent  the  algebraic  sum  of  three  functions  of 
Xj  the  equation  may  be  written 

y  =  u-\-v  —  w.  (1) 

If  an  increment  Ax  is  given  to  x,  the  variables  y,  u,  v  and  Wy  which 
are  functions  of  x,  will  take  the  corresponding  increments  Ay,  Ati,  Av 
and  Aw,  respectively ;  then  (1)  becomes 

y  -\-  Ay  =  (u  -\-  Au)  -{-  (v  +  Av)  —  (w  +  Aw).  (2) 

Subtracting  (1)  from  (2), 

Ay  =  Au  -{-  Av  —  Aw.  (3) 

Dividing  by  Ac.,  Ay  ^  A«  ^  A^  _  A«,_ 

Ax      Ax      Ax       Ax 

When  Ax  approaches  zero, 

limit  of  ^  =  ^,  limit  of  ^  =  ^,  etc.,  by  Art.  7. 
Ax     dx  Ax     dx 

*  In  this  definition  of  an  algebraic  function  it  is  understood  that  the  operations 
are  not  repeated  an  infinite  number  of  times. 

11 


12  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Therefore  ^  =  ^  +  ^  _  ^,  by  Art.  5,  1st  and  2d ; 

ax     ax     ax      ax 


(fB 


d(u-\-v  —  w)_dudv     dw  ^^    i 

dx  dx     dx     dx  .  VJ-^ 


If- the  algebraic  sum  of  four  or  more  variables  be  given,  the  deriva- 
tive would  be  found  similarly. 

I.  Hence,  the  derivative  of  the  algebraic  sum  of  any  number  of  func- 
tions of  X  is  equal  to  the  algebraic  sum  of  their  derivatives. 

Art.  11.   Product  of  a  Constant  and  a  Function. 

Let  a  represent  any  constant,  then  the  product  of  a  constant  and 
a  function  of  x  may  be  written 

y  =  av.  (1) 

Let  V  and  y  take  the  increments  Av  and  ^y  corresponding  to  the 
increment  Aa;  given  to  x,  then 

y-\-Ay  =  a(v  +  Av).  (2) 

Subtracting  (1)  from  (2),  Ay  =  aAv. 

Dividing  by  Aa;,  _ ^=a  — •  (3) 


When  Aa;  approaches  zero,  by  Art.  5,  1st, 

limit^  =  limitfa^^; 
Aa;  V    ^^J 

therefore  ^  =  a  ^,  by  Art.  7. 

dx        dx 

If  v  =  x,  Av  =  Aa;  and  -^  =  a. 
dx 


6 


II.  Hence,  the  derivative  of  the  product  of  a  constant  and  a  f  motion 
of  X  is  the  product  of  the  constant  and  the  derivative  of  the  variable. 

Art.  12.     Any  Constant. 

As  the  value  of  a  constant  remains  unchanged  in  any  one  discus- 
sion, the  constant  receives  no  increment,  or,  in  other  words,  the  incre- 
ment of  the  constant  is  zero. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS.  13 

Let  a  represent  any  constant ;  then 


Aa  =  0    and    ~  =  0. 

Ax 

Therefore,  when  Ax  approaches  zero, 

dx 
III.   Hence,  the  derivative  of  a  constant  is  zero. 


o 


Art.  13.     Product  of  Two  Functions. 

Let  the  product  of  two  functions  of  x  be  represented  hy  y  =  uv. 
When  X  is  given  an  increment,  the  variables  v,  u  and  y  receive  corre- 
sponding increments,  and  the  equation  becomes 

y  -^  Ay=  (u  +  Au)  (v  +  Av) 

=  uv  -\-  uAv  -\-  vAu  +  Au  Av,  (1) 

Hence  Ay=(v-{-  Av)  Au  +  uAv ;  "^         (2) 

and  ^  =  (v  +  ^v)^  +  u^.  (3) 

Ax  Aic         Ace 

When  Ace  approaches  zero, 

limit  ^  =  !^,  limit  «^  =  «*!, 

Aa;     dx  Ax        dx 

limit  (v  +  Av)  =  v,    and    limit    ^  =  ^. 

Ace      dx 

Therefore,  by  Art.  5, 

dy  _d  (uv)  _    diu        d/v                              /^TvJ 
dx        dx  dx        dx  v j/ 

IV.  Hence,  the  derivative  of  the  product  of  two  functions  of  x  is  the 
sum  of  the  products  of  each  function  by  the  derivative  of  the  other. 

Art.  14.     Product  of  Three  or  More  Functions. 

Let  the  product  of  three  functions  of  x  be  represented  hj  y  =  uvw. 
The  product  of  two  of  the  functions,  as  uv,  may  be  taken  equal  to 
z ;  then,  by  the  preceding  article, 


14  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

d(uvw)      d(zw)         dz   ,     dw 
.,    ,  dx  dx  dx       dx 

*     ^   Z  r  ac  \  dx        dx)         dx 

*  du  ,        dv  ,        dw 

=  wv \-ivu ^uv • 

dx  dx  dx 


G 


This  process  may  be  extended  to  the  differentiation  of  the  product 
of  any  number  of  functions. 

V.  Hence,  the  derivative  of  the  product  of  any  number  of  functions 
of  X  is  equal  to  the  sum  of  the  products  of  the  derivative  of  each  'into 
the  product  of  all  the  others. 

Art.  15.     Quotient  of  Two  Fuxctioxs. 
Let  the  quotient  of  two  functions  of  x  be  represented  by  2/  =  — 


Then 

vy  =  u; 

and,  by  V., 

dx       dx     dx 

Therefore 

du        dv 
dy     dx        dx 

dx 
*  dx        dx  /!      \ 

= — ^ —  (>i.) 

VI.  Hence,  the  derivative  of  a  fraction  is  equal  to  the  denominator 
multiplied  by  the  derivative  of  the  numerator,  minus  the  numerator  multi- 
plied by  the  derivative  of  the  denominator,  divided  by  the  square  of  the 
denominator. 

du 
CoR.  1.   If  the   numerator   is   constant,   —  =  0,  by   III.,  and  VI. 

dx 

becomes  „.  dv 

u  — 

dy_        dx 
dx~       v^ 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS. 


15 


Hence,  the  derivative  of  a  fraction  with  a  constant  numerator  is  ( 
negative  and  equal  to  the  numerator  multiplied  by  the" derivative  of  j 
the  denominator,  divided  by  the  square  of  the  denominator. 

CoR.  2.   If  the   denominator  is   constant,   —  =  0,  by  III.,  and  VI. 

dx 


becomes 


dy  ^ 

dx       v^ 


du     du 
dx      dx 


which  is  the  same  result  that  would  be  obtained  by  II. 


Art.  16.     Constant  Power  of  a  Function. 

Case  1.   When  the  exponent  is  a  positive  integer. 
Let  -y  be  a  function  of  x,  and  n  its  exponent ;  then 

y  =  V", 
2/  +  A?/  =  (v  +  Av)", 
and  Ay  =  (iJ  H-  A?;)"  —  v". 

Expanding  (y  -\-  Av)",  by  the  Binomial  Theorem,  and  dividing  by  Aa, 


Ay 
Ax 


-^v"-2(Av)  ...  +{^vY  ' 

Li 


A'y 
Aa; 


When  Ao;  approaches  zero,  Av  approaches  zero  also;  hence 


d\i         „_i  dv 
dx  dx 

Case  2.   When  the  exponent  is  a  positive  fraction, 

m 

Let  y  =  v''^        (t/ 

then  ?/**  =  v"". 

As  m  and  n  are  positive  integers,  by  Case  1, 

dx  dx 


therefore 


dy_m  v"'^^  dv  _m  i;"*'^  d^ 
dx     n  v""^  dx      n    m--  dx 

^m^-^dv^ 
n         dx 


VIL)  ^- 


5»w 


*•  r<.( 


«i 


«.r 


•1-1  -  M  vVir-.;.  ^-.  ivr,.]*" 


Ui 


10  DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 

Case  3.  When  the  exponent  is  negative  and  either  integral  or 
fractional,  as  —  n. 

Let  y  =  v~", 

then  y  =  — 

Differentiating  by  Art.  15,  Cor.  1, 

da;  'y'"  dx  •        ^ 

VIT.  The  derivative  of  a  constant  power  of  a  function  of  x  is  equal  to 
the  product  of  the  exponent,  the  function  ivith  its  exponent  diminished  by 
unity,  and  the  derivative  of  the  function. 

Radical  expressions  may  be  differentiated  according  to  this  rule, 
the  quantities  being  first  transformed  into  equivalent  expressions  with 
fractional  exponents. 

The  radical  of  the  second  order  is  the  one  that  occurs  most  fre- 
quently.    It  is  differentiated  as  follows  : 

Let  y  =  Vv  =  v2. 

Hence,  the  derivative  of  the  square  root  of  a  function  of  x  is  equal 
to  the  derivative  of  the  function  divided  by  twice  the  square  root  of 
the  function. 

PROBLEMS. 

The   formulas   established  in   this  chapter   are   sufficient  for  the 
differentiation  of  all  algebraic  functions  of  a  single  variable. 
Differentiate  the  following  functions : 

1.   y  z=z  a  +  bx  ■{■  :x?. 

|=d>  +  ^-^^) 

^^^IM  +  ll^,  byL 
dx        dx  dx  ^ 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS.  17 

^  =  0,  by  III. ;  ^-^^  =  6,  by  11. ;  ^^  =  3a^,  by  VII. 
dx  dx  dx 

Therefore  ^=5  +  3ar^. 
dx 

2.  y=  (a+x)(b-{-2af). 

^=(b  +  20^ %t^)  +  (a  +  a;)^(^±2^,  by  IV., 
dx  dx  dx 

=  (b-\-2x^-^4:(a-\-x)x 
=  b-{-6x-  +  Aax. 

3.  V  = • 

^      (a  +  x'f 

Applying  VI.,  and  VII., 

dy_{a  +  a^yx  12o^  -  4:X^  x  S(a -\- x'Y  x  (2x) 
dx  (a  +  i^y 

^12x^(a-x^ 
(a  +  ary 

4.  y  =  x(l  +  x'){l-^o^). 

^^=(1  -\-ar^(l  ^  :^)  +  x(l  +  af)£(l  +a^  +  x(l  -\-x^)£(l  +x')      . 
=  (1  +  x2)(l  +  ar3)+  a^(l  +  a^)(2a')+  a;(l  +  a:2)(3a?2) 

\-\-x  dy  _  1  —  2x  —  a^ 


5.    y  = 


l+x"  dx         (l-^-x'f 


r-  dy        a   ■ 

7. .  ?/  =  (a+x)'"(6+a;)".     ^  =  (a  +  a;)'"-^(6  +  x)'*-i[m(6  +«?)  +  w(a  +  a;)]. 


9.    2/=-^^ 


^  =  _  JL.. 

dy  _  _    g  +  3x 
(?a:~     2VaT^' 


10.   yz=  (a  —  x)  Va  +  a:. 

0 


18  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

/  2  .     o^    /-2 -2  -       cly_a'-{-a^a:^-4:X* 

11.    y  =  x(a^-^x-)Va^  —  af.  -^= ,  

if  \     ^     )  clx  -yja'-y? 


12.    2/  = 


dy_  1 


•Vl-a^  ^^^     ■yJl-x^  +  2x{X-x') 


13.    y^Vl  +  a^  +  Vl-g;.  'dy^      l-f-Vi-a;^ 


dx 

3a^  +  2  dy^  2 

16.  ^^vTTZ±V^^.  ^  =  _2^^ 


14.  y=(l-Zx'-\-Q>x%l  +  x')\         ^  =  60a^(l  +  £c2)2^ 

O  ^    I     o 

15.  y  = 


3  (aj2  _^  1)1(4  a^  _  3).  ^  =  56  cc3(a^  +  l)i 

clx 

2a;^-l  ^  =  J_±i^. 

a;  Vr+¥2  da;     ^2(-l  _|_  ^,2)1 


1 Q     .,  —  V(^  +  c^)^  ^  _  (a?  —  2  g)  Va;  +  g 

ly.    ?/ —        •  —  g         • 

Va;  — g  "-'^  (a;_ay2 

20     ^^        ^  ^  (Zy^      m(6  +  a;)+n(a  +  a?). 

(g  +  a;)*"  (6  4-  a;)"  da:  (g  +  xY-^\b  +  a;)"+i 


21.     y=( E Y 


cZy_        ny 


22     .  =  ^/-iEZ.  dy_  2a:(2-a^ 

\(l  +  a;2)«  da;  (X-x^)l(l^^)^ 


CHAPTER   III. 

DIFFERENTIATION  OF  TRANSCENDENTAL  FUNCTIONS. 

Art.  17.     Definitions. 

All  functions  that  are  not  algebraic  are  called  transcendental.  Tran- 
scendental functions  are  divided  into  four  classes : 

1st.  Logarithmic  functions ;  those  in  which  a  logarithm  of  a  variar 
ble  is  involved. 

2d.  Exponential  functions ;  those  in  which  a  variable  enters  as  an 
exponent. 

3d.  Trigonometric  functions;  those  involving  sines,  cosines,  tan- 
gents, etc.,  in  which  the  arc  is  the  independent  variable. 

4th.  Inverse  trigonometric  functions;  those  derived  from  trigono- 
metric functions,  by  taking  the  arc  as  the  dependent  variable.  Thus, 
from  the  trigonometric  function,  y  =  sin  x,  is  obtained  the  inverse 
function,  x  =  arc  sin  y,  which  is  read,  "  x  equals  the  arc  whose  sine  is 
2/."  The  inverse  trigonometric  functions  are  also  called  circular  func- 
tions and  anti-trigonometric  functions. 

The  inverse  trigonometric  functions  are  often  expressed  differently, 
as  shown  in  the  following  identities : 

arc  sin?/  =  sin^^ y ;    arc  tan y  =  tan "^2/ 5    ^^^  cosec y  =  cosec"^?/. 

This  second  notation,  employed  to  express  inverse  trigonometric  func- 
tions, was  suggested  by  the  use  of  negative  exponents  in  algebra,  but 
the  student  is  cautioned  against  the  error  of  regarding  sin~^2/  ^^  equiva- 
lent to  -: Another  application  of  this  notation  for  inverse  func- 

smy 

tions  is  seen  in  an  anti-logarithm;  ii  y  =  logo;,  then  x  =  log'^y. 

19 


20  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

Art.  18.     Base  of  the  Natukal  System  of  Logarithms. 
The    base    of    the   natural  system  of  logarithms  is  the  limit  of 
fl;-f-J  as  X  approaches  infinity. 

By^-the  Binomial  Theorem, 

=  1-1-1  4 U  .^^ i-^ L  -4-  .... 

^     ^  1.2  ^         1.2.3  ^ 

When  X  increases  indefinitely,  , 

This  limit  is  usually  denoted  by  e. 

Therefore  e  =  H-  1  +  r^^  +  —i-^  +  -• 

By  summing  this  series  the  value  of  e  is  found  to  be  2.7182818+, 
which  is  the  base  of  the  natural  system  of  logarithms. 

Art.  19.     Logarithmic  Functions. 

Throughout  this  chapter,  v  and  u  will  always  be  regarded  as  func- 
tions of  a  single  independent  variable  x. 

Let  the  base  of  the  system  of  logarithms  be  a ; 

then  let  y  —  log,  v ; 

hence  y  +  A^  =  log,  {v  +  Av), 

A2/  =  log,  {v  +  Av)  —  log,  V 


V  \  V  J 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS.    21 


^=^iogji+— y°- 


Av 

and  — ^  =  —     ^»  ■  . 

Ax       V  \         V  J 

Now  as   Ax.  approaches  zero,  Av  approaches  zero;  and  therefore 
—  approaches  zero  and  —  approaches  infinity. 

If  —  be  substituted  for  x  in  the  preceding  article,  the  limit  of 


(-?) 


IS  seen  to  be  e. 


dv 


Therefore  ^=log,e^.  YIII. 

clx  V 

Log„e  is  the  modulus  of  the  system  in  which  the  logarithm  is 
taken  and  may  be  denoted  by  M. 

VIII.  Hence,  the  derivative  of  the  logarithm  of  a  function  of  x  is 
equal  to  the  modidus,  multiplied  by  the  derivative  of  the  function,  divided 
by  the  function. 

Hereafter,  when  no  base  is  specified,  it  will  be  understood  that 
natural  logarithms  are  used ;  then 

Jbr=log«e  =  log,e  =  l, 

aud  VIII.  becomes  ^  =  — -^  VIIL  a. 

dx     dxv      ■ 

Art.  20.     The  Exponential  Function  with  a  Constant  Base. 

Let  the  exponential  function  with  a  constant  base  be 

y  =  a\ 
Taking  the  logarithm  of  each  member, 

log  y=vlog  a. 

Differentiating  by  VIII., 

dy 

,^dx     ,        dv 
M —  =  loga3-; 


22  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

IX. 

dx 

Ajid  when  natural  logarithms  are  used, 


therefore  dy^a^gaclu_ 

dx         M     dx 


•:  f?  =  „.logaf!.  IX.a. 

dx  dx 

If  a  =  e  in  IX.  a,  since  log^e  =  1, 

iiL^e^.  IX.  6. 

dx         dx 

li  v  =  xm  IX.  a,  ^^  =  a^  log  a.  (1) 

Ifa  =  ein(l),  ^  =  e^  IX.  c. 

dx 

IX.  Hence,  ^^e  derivative  of  an  exponential  function  ivith  a  constant 
l)ase  is  equal  to  the  function  multiplied  by  the  logarithm  of  the  base  and 
by  the  derivative  of  the  exponent,  divided  by  the  inodidus. 


Art.  21.     The  Expoxential  Function  with  a  Variable  Base. 

Let  the  exponential  function  with  a  variable  base  be 

y  =  u\ 
Then  log  y  =  v\o%u] 

dy  du 

-^  V  — 

and  by  VIII.,  M —  =  M— h  log  u  —' 

y  u  ax 

mi,      £  dy         „_,  du  ,  u"  log  u  dv  v 

Therefore  -f-  =  vu"^  —-  H r-^ — -  •  X. 

dx  dx         M     dx 

X.  Hence,  the  derivative  of  an  exponential  function  with  a  variable 
base  is  equal  to  the  sum  of  tioo  derivatives ;  the  first  being  obtained  as 
though  the  base  ivere  variable  and  the  exponent  constant,  and  the  second 
as  though  the  base  were  constant  and  the  exponent  variable. 


DIFFERENTIATION   OF   TRANSCENDENTAL  FUNCTIONS.     23 

PROBLEMS. 

1.  y=ia:^\ogX. 

2.  ?/ =  log (2 a;  +  o^. 
a-\-x 


3.   2/  =  log 


a  —  x 


4.  2,  =  logV|g- 


5.  2/  =  log(^  + Vl  H-ar^. 

6.  2/  =  loga^. 

7.  2/  =  log^^' 

8.  y=  log  (logic). 


,      -Vx^  -\-l—x 
9.    ^  =  log-— ^= 

VaH  H-  1  +  aj 


10.    2/  =  l0g--=^r^^— ==:• 

Vl  4-  »— Vl  —a; 


11.  2/  =  log(VH-a)'+Vl-ar). 

12.  2/  =  a**- 

13.  ?/  =  «''•  ^  =  2a''.loga.a?. 


14.   ?/  =  e*(a;  — 1). 


da; 

:  a;  (2  log  a;  4-1). 

c/2/_ 

24-3ar^ 

ca- 

2a;  +  a^ 

cly_ 

2a 

dx 

a'-x" 

dy_ 

1 

dx 

1-ar^ 

dx 

1 

Vl4-ar^ 

dy_ 
dx 

_2^ 
x 

^V- 

_21oga; 

dx 

x 

dy 

1 

dx 

X  log  X 

dy_ 

2 

dx 

Va;2  4-1 

dy 

1 

dx 

x^l-^ 

dy  _ 

1             1 

dx- 

^     X  VI  -  a;^ 

dy_ 
dx 

=  a'^'e^loga. 

dx 
dx 


15.   2/  =  2eV5(a;'^-3a;4-6a;^-6).  ^  =  a;eVi. 

Ota; 


24  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


'      e^  +  1 

K. 

V-    '' 

^      1+a; 

18. 

•-       1 

2/  =  af. 

19. 

y  =  af  a*. 

20. 

i/  =  af. 

21. 

2/  =  a;^^^''.  ■ 

22. 

y  =  yf. 

dx 

{e  +  lf 

dy 

xe 

dx 

(!  +  »;)» 

dy_ 

.af(l- logic) 

da; 

=  a^^a;'*-^ 

(n  4-  a;  log  a), 

da; 

=  af  (loc 

^a;  +  l). 

dy_ 
dx 

=  log  a.-^ 

,  ^logx-l 

^^  =  af"  Aog  a;  +  log^  x  +  ^V. 


23.    y  =  e'[xr—nyf-'^+n{n-l)x^-^ ].     ^  =  e=»a;^ 

cta; 


Art.  22.     Circular  Measure.* 

In  higher  mathematics,  angles  are  not  measured  by  the  ordinary 
degree  or  gradual  system,  but  in  terms  of  another  unit.  The  circular 
measure  of  an  arc  of  a  circle  is  the  ratio  of  the  length  of  the  arc  to  the 
length  of  its  radius ;  and  it  is  evident  that  this  ratio  does  not  vary 
with  the  radius.     Thus  the  value  of  an  arc  of  360°  in  circular  measure 

is  ^=  27r,  of  180°  is  IT,  of  90°  is  J,  and  of  1°  is  ~ 
r  2  180 

The  angle  at  the  centre  of  a  circle  subtended  by  an  arc  equal  to  the 
radius  is  the  radian  or  circular  unit. 

Let  X  denote  the  number  of  degrees  in  an  angle,  and  z  the  number 
of  radians  in  the  same  angle;  then  since  there  are  tt  radians  in  two 

right  angles, 

X    _  z 
180  ~  7 

Therefore  z  =  -^  x, 

180 

*  Hall's  Mensuratioiu  §§  9,  10,  11,  and  12. 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS.    25 


and 


180 

x= z. 

IT 


Hence,  to  reduce  from  gradual  to  circular  measure,  the  number  of 


degrees  in  the  angle  is  multiplied  by 


180 


and  to  reduce  from  circular 


to  gradual  measure,  the  circular  value  is  multiplied  by 


180 


Art.  23.     Limiting  Value  of 


SIX  ^ 


Let  the  small  angle  AOB  in  Fig.  4  be  represented  by  $,  and  the 
radius  OA  by  a;  and  let  BC,  AB  and  AD  be  sin^,  chords  and  tand, 
respectively. 


C  A 


Fig.  4. 


The  area  of  the  triangle  AOB  =  \a^  sin^; 
The  area  of  the  sector     AOB  =  \a^d-, 
The  area  of  the  triangle  AOD  =  \o?  tan^; 

and  these  areas  are  obviously  in  an  ascending  order  of  magnitude ; 
hence  tan  ^  >  ^  >  sin  ^, 

tan^ 


or 


6 


Thus  - — -  lies  between    . 

sm  6  sm  6 


sin  6     sin  6 
tan^ 


>1. 


and  1;  but  when  0  approaches  zero, 

e 


-1^  or  approaches  1 ;  hence,  as  6  diminishes  indefinitely,  -^ — - 

sind       cos^    FF  >  J  j^  g.^^ 


approaches  the  limit  unity. 


26  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

Art.  24.     Trigonometric  Functions. 

1.   Differentiation  of  sin  v. 

Lfet  y  =  sin  v, 

then  y-}-Ay  =  sin  (v  +  Av)  ; 

therefore  Ay  =  sin  (v  -f  Ai?)  —  sin  v. 

By  Trigonometry, 

sin.4  -  sin5=  2  cos|-(^  +  5)  sin^  (.4  -  B). 

Substituting  v  -^  Av  =  A,  and  v  =  B,  in  this  formula, 

A3/  =  2cosfv+— jsin  — ; 

sm 
hence  —  =  cos  (  v  -{-- — 


Ax  \         2  J    ^     Ax 

2 

When  Aflj  approaches  zero,  Av  approaches  zero,  and  by  Art.  23, 

.    Av 
sm  — 

limit is  unity. 

Ay  ^ 

2 

Therefore  ^=cos'y^.  XI. 

ax  ax 

XL   Hence,  the  derivative  of  sin  v  is  equal  to  cos  v  multiplied  by  — • 

dx 

2.  Differentiation  of  cos  v. 

Let  y  =  cosVf' 

then  Ay  =  cos  (v  +  Av)  —  cos  v. 

By  Trigonometry, 

cos  ^  -  cos  5  =  -  2  sin  ^(A-{-B)  sin  ^(A-B). 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS.     27 
Substituting  v  -\-  Av  =  A,  and  v  =  B, 

A2/  =  -2sin(.  +  f)sinf; 

sin^ 

hence  ^^-smfv^^] ^^. 

Ax  \        2  J    Av    Ax 

2 

Therefore  ^  =  _  sin  v — .  XII. 

dx  dx 

XII.   Hence,  the  derivative  of  cosv  is  negative^  and  equal  to  sinv 

multiplied  by  —  • 
dx 


Let  y  =  tan  v 


3.  Differentiation  of  tan  v. 

sinv 
cosv 

cos  V  —  (sin  v)  —  sin  V — (cos  v) 

o   dv  ,     •  o  dv 
cos^v hsm^v- 


dx\QosvJ  cos'^v 

dv  ,     •  2  dv 
cZa;  die 


cos^v 

=  sec'v—'  XIIL 

da; 

XIII.   Hence,  the  derivative  of  tanv  is  equal  to  se(^v  multiplied 
4.  Differentiation  of  cotan  v. 

•r      .  .  cosv 

Let  2/  =  cotan  v  = 

smv 


.      f       .      dv\  f         dv\ 

smvf  —  sm  V-—   —  cosv  (cosv —  ) 

By  VI.  ^  /cosv\  ^         V ^ V         ^^J 

*'  dx\mivj  sin^v 


=  -cosec2v— .  XIV. 

dx 

XIV.   Hence,  the  derivative  of  cotan  v  is  negative^   and  equal  to 
cosec^v  multiplied  by 


dx 


By  Art.  15,  Cor.  1, 


28  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

5.  Differentiation  of  sec  v. 

,    Let  y  =  secv  = • 

♦.  cos  V 

d  /        s        •      dv 
—  (cos-y)     sinv — 
dy^d  r   1    \^     dx^        ^_ dx 

dx     dx\GosvJ  cos^v     "~   cos^v 

=  sec  V  tan  v  — •  XV. 

dx 

XV.   Hence,  the  derivative  of  sec  v  is  equal  to  sec  v,  multiplied  by 

dv 

tanv,  into  — • 
dx 

6.  Differentiation  of  cosec  v. 
Let  y  =  cosec  v 
By  Art.  15,  Cor.  1, 


sm  V 


dy^  df  1   \ 


) 


dv 

cos?; — 

dx 


dx     dx\smvj  sm^y 

=  —  cosec  V  cotan  v—-  XVI. 

dx 

XVI.  Hence,  the  derivative  of  cosec  v  is  negative^  and  equal  to  cosec  Vj 

dv 
multiplied  by  cotan  v,  into  — 

dx 

7.  Differentiation  of  vers  v. 

Let  y  =  vers  v  =  l  —  cos  v. 

Then  ^  =  —  (1  -  cos  v)  =  sin  v  ^.  XVII. 

dx     dx  dx 

XVII.  Hence,  the  denvative  of  vers  v  is  equal  to  sin  v  into  — • 

'  dx 

8.  Differentiation  of  covers  v. 

Let  y  =  covers  v  =  l  —  sin  v. 

Then  ^  =  —  (1  -  sin  v)  =  -  cos  v  — •  XVIII. 

dx     dx  dx 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS.    29 

XVIII.   Hence,  the  derivative  of  covers  v  is  negative^  and  equal  to 

.  .    dv 

cos  V  into  — 

dx 

Art.  25.     Inverse  Trigonometric  Functions. 

It  should  be  remembered  that  there  are  two  ways  of  indicating 
the  inverse  trigonometric  functions.  The  functions  arc  sin  x,  arc  cos  x, 
arc  tan  ic,  etc.,  are  often  written  as  follows:  sin~^ic,  cos~^a;,  tan  ""■'a;,  etc., 
respectively. 

1.   Differentiation  oi  y  =■  arc  sin  v. 

Then  v  =  sin  y. 

By  XL,  T  =  "^'yT-^ 

dx  dx 


hence 


dy  _    1    dv  __  1  dv 

dx     cos  y  dx     Vl  —  sin^w  ^^ 

dv 


Therefore      ^  (arc  sin  ^)^      dx     ^    . 
dx  VI  —v^ 

2.   Differentiation  of  y  =  arc  cos  v. 

Then  v  =  cos  y. 

ByXIL,  ^  =  -smy^', 

^         '  dx  ^  dx' 


hence 


dy  _        1    dv 1  dv 

dx         sin  y  dx         Vl  —  cos^y  ^^' 
dv 


XIX. 


Therefore      ^(^rccos.;)^ dx__^  ,  ^x. 

dx  Vl-'y^ 

3.   Differentiation  of  y  =  arc  tan  v. 
Then  v  =  tan?/. 

hence  ^  =  _i_^=- 1 ^. 

c?a;     sec^  ?/  (^x     1  +  tan^  y  dx 


30  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


dv 

Therefore 

d  (arc  tan  v)         dx 
dx         ~l+v''' 

4.   Differentiation  oi  y  —  arc  cot  v. 

Then 

V  =coty. 

By  XIV., 

dv                  9    dv 
—  =  -cosec2  2/-^; 
dx                   "^  dx 

lence 

dy  _           1      dv  _              1          dv 

dx         cosec^  ydx         1  +  cotan^  y  dx 

dv 

Therefore 

d  (arc  cot  v)             dx 

XXI. 


-     ,   ,     o-  XXIL 

dx  1  -f  v^ 

5.  Differentiation  ofy  =  arc  sec  v. 
Then  v  =  sec  y. 

By  XV.,  ^=  sec ^  tan 2/^; 

da;  da; 

hence  dy^        1        di;^  1  dv^ 

dx     secytB,nydx     sec?/ Vsec^y  -  l^^a; 

Therefore  djavcseov)  ^       ^      .  XXIII. 

da;  i;  V-y*^  -  1 

6.  Differentiation  ot  y  =  arc  cosec  v. 
Then  v  =  cosec  y. 

By  XVL,  ^=  -  cosecycoty^; 

dx  dx 

hence  ^  =  -_-J_^  = _^____^. 

da;         cosec  2/ cot  2/ da;         cosec  ?/ Vcosec^y  -  1  ^« 

dv 
Therefore  cZ  (arc  cosec  v)  ^  _       da;      ^  ^-j^j^ 

da?  V  Vv^  —  1 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS.    31 

7.   Differentiation  of  y  =  arc  vers  v. 
Then  v  =  vers  y. 


By  XVII., 
ice              ^  = 

dx             dx                         dx 

1          dv                    1                 dv 

dx 
Therefore 

Vl  —  cos^i/  ^^'      V2  vers  y  —  veis^y^^ 
do 
d  (arc  vers  v)  _        dx 

^ XXV. 

dx  V2  V  —  v^ 

8.    Differentiation  oi  y  =  arc  covers  v. 
Then  v  =  covers  y. 

ByXVIIL,       *'  =  -cos2/^=-Vl-sin'2,^; 
dx  dx  dx 

hence  ^  = ^  ^= ^  — ^. 

da;         Vl  —  sin^?/  ^^         V2  covers  y  —  covers^  ?/  ^^^ 

Therefore         d  (arc  covers  7>)  ^ -Jf XXVL 

da;  \/2  V  —  v^ 


PROBLEMS. 

1 .  2/  =  sin  na;.  -^=n  cos  nx\ 

da; 

2.  ?/  =  sin*'a;.  -^=  nsin^^^ajcosa;. 
^  da; 

3.  2/  =  cot2(a;»).  ^  =  -  6  a;^ cot  (a.-^)  cosec^ (a;^) 

4.  2/  =  logCsin^a;).  -^=2  cot  a;. 

da; 

6.   2/  =  sin  2  a;  cos  a;.  -^  =  2  cos  2  a;  cos  a;— sin  2  a;  sin  a;. 

dx 

6.   2/  =  e*cosa;.  -^  =  e='(cosa;  —  sina;). 

da; 


32  DIFFERExXTIAL   AND   INTEGRAL   CALCULUS. 

7.   j/=e««^^sina;.  ^^=  e'^'' (cosx  -  sin^x). 

clx  ^ 

«.   y  =  smlogx.  ^  =  icos(loga;). 

clx      X       ^    °    ^ 

9.   i/=  (cosa;)"°^  -^=  (cosa;)"°*[cosa;logcosa;  -  sinajtana;]. 

10.    y  =  logt3inx.  ^ 


11.  y  =  logJ\±^. 

^  1  —  Sin  a; 

12.  y  =  logsecx. 


13.   2/ =  arc  sin-, 
a 


14.  2/ =  arc  tan?. 

a 

1  — a^ 

15.  2/  =  arc  sin 

16.  ?/  =  arcsin(3a;  — 4a:^. 

17.  2/ =  arc  sec  2a?. 
2x 


18.    ?/  =  arctan 


1-ar^ 


19.   y  =  x  Va-  —  a;^  +  a^ arc  sin  -• 


20.    2/  =  6^ 


dx 

sin2aj 

dx 

1 

cos  a; 

dx 

=  tana;. 

dy_ 

1 

dx 

Va^-ar' 

dy_ 

a 

dx 

a^^y? 

dy_ 

2 

dx 

1-hx^ 

dy^ 

3 

: -. 

dx 

VI -x' 

d^_ 

1 

dx 

xV^x^-1 

dy^ 
dx 

dy_ 
dx 

2 

l+a;^ 

:2Va=^-a;^. 

dy_ 

garctanx 

clx     l+r' 


21.  2,  =  V^-r^  +  aarosm--  _  =  (^__j. 

22.  2,  =  arc  tan  ^^:^^.  ^  =  1. 

VI  +  COS  a;  dx     2 


DIFFERENTIATION  OF   TRANSCENDENTAL  FUNCTIONS. 


23.  3/  =  arccot«  +  logJ^.  dy_2a^ 

X  ^x  -\-  a 

„ .  X  (arc  sin  ic)  ,  ,        /^ -s 

24.  y=    ^  ^  +  log  VI  —  ar. 

25.  2/  =  ilog(^^'  +  -^arctan2.^. 

26.  y^^"^(^sina^-cosa;)^  ^  =  e-siiia.'. 

a^  + 1  dx 

a;V5  di/  1 

27.  2/  =  arc  sec -^  — 


dx 

x'- 

a* 

dy_ 

arc  sin  x 

dx 

(1- 

-.^)t 

dy  _ 

1 

28.   2/  =  log\/—i_^  +  ^  arc  tana?. 
^1  —  X 


2Var^  +  a;-l  ^^     x^x^  +  x-1 


dy^     1 
daj     1  —  a;* 


•:  CHAPTER   IV. 

DIFFERENTIALS. 

Art.    26.     Introduction. 

The  formulas  for  differentiation  given  in  the  preceding  chapters 
have  been  established  by  the  method  of  limits.  In  this  chapter  another 
method  of  treatment  will  be  presented  which  is  called  the  method  of 
infinitesimals.  According  to  this  second  method,  the  independent  varia- 
ble is  supposed  to  change  by  the  continued  addition  of  an  infinitely 
small  constant  increment.  This  increment  is  called  the  differential  of 
the  variable,  and  the  corresponding  increment  of  the  function  is  called 
the  differential  of  the  function.  The  differential  of  a  variable  may 
then  be  defined  as  the  difference  between  two  consecutive  values  of  the 

variable.     Hitherto,  the  symbol  -^  has  been  regarded  as  a  whole,  but 

dx 

here  it  is  defined  as  the  ratio  of  the  differential  of  the  function  to  the 
differential  of  the  independent  variable,  and  is  regarded  as  a  fraction. 
The  phraseology  and  notation  of  the  two  methods  are  different,  but 
they  give  identical  results.     To  illustrate : 

Let  y=^, 

thenbyVIL,  ^  =  5x\ 

dx 

If  differentials  are  used,  the  equation  becomes 

dy  =  5  x*dx, 

which  would  be  read,  "The  differential  of  y  is  equal  to  5x^  times  the 
differential  of  x." 

In  general,  let  ^  =  /'(ic), 

dx 

then  dy  =  f'(x)  dx. 

34 


DIFFERENTIALS. 


35 


Now  the  reason  for  sometimes  calling  the  derivative  the  differential 
coefficient  is  apparent,  as  it  is  seen  to  be  the  coefficient  of  dx  in  the 
differential  of  f(x). 

If  each  member  of  each  of  the  formulas,  I.-XXVI.,  be  multiplied 
by  dx,  a  corresponding  set  of  formulas  will  be  obtained  for  the  differ- 
entials of  functions.* 


Art.   27.     Geometric  Interpretation  of  -^• 

dx 

The  two  methods  may  be  compared  geometrically.     In  Fig.  5,  let 

AB  represent  any  plane  curve  whose  equation  will  show  the  relation 

between  the  coordinates  of  any  point  of  the  curve ;  then  the  ordinate 

y  may  be  expressed  as  a  function  of  x,  giving  for  the  equation 

y  =  f(.^)- 
1.   By  the  method  of  limits. 

Let  (x,  y)  be  the  coordinates  of  any  point  P  of  the  curve   and 
{x  +  Aaj,  y  +  A?/)  the  coordinates  of  any  second  point  P%  and  i/^  the 


Fig.  5. 

angle  which  the  tangent  to  the  curve  at  P  makes  with  the  X-axis.  If 
6  be  the  angle  which  the  secant  through  P  and  P'  makes  with  the 
X-axis,  and  PM  be  drawn  parallel  to  OX, 

*  Lagrange  in  Mecanique  Annlytique  says  :  "  When  we  have  properly  con- 
ceived the  spirit  of  the  infinitesimal  method,  and  are  convinced  of  the  exactness  of 


2§  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

tan^  =  ^=^. 
PM     Ax 

^  Now,  suppose  that  P'  approaches  P,  or,  in  other  words,  that  Ax 
decreases  toward  zero ;  evidently  0  approaches  ip,  and  tan  0  approaches 
tan  j/^  as  its  limit. 

By  definition,  limit  ^  =  ^. 

Ax     dx 

Therefore,  by  Art.  5,  tan  ^  =  -^. 

•^  '  ^      dx 

2.   By  the  method  of  infinitesimals. 

Let  P  and  P'  represent  consecutive  points  on  the  curve,  then  PN 
and  P'N'  are  consecutive  ordinates.  The  part  of  the  curve,  PF,  called 
an  element  of  the  curve,  is  regarded  as  a  straight  line,  and  when  pro- 
longed it  forms  the  tangent  to  the  curve  at  the  point  P.  PM  is  drawn 
parallel  to  OX; 

then  PM=NN'=dx, 

and  MP'  =  dy. 

And  as  .,  ^  =  ^, 

^      PM     dx 

Hence,  the  derivative  of  the  ordinate  at  any  point  of  a  plane  curve 
with  respect  to  the  abscissa  is  equal  to  the  tangent  of  the  angle  which 
the  tangent  to  the  curve  at  that  point  makes  with  the  X-axis.* 

Art.  28.     Geometric  Derivation  of  the  Formulas  for  the 
Differentiation  of  the  Trigonometric  Functions. 

In  Fig.  6,  let  AP  represent  a  circular  arc  x,  with  radius  =  1,  and 
PP'  =  dx  an  infinitely  small  increment  given  to  x.  PS  is  drawn  par- 
allel to  OA,  and  PN  and  P'M  are  consecutive  ordinates. 

the  results  by  the  geometrical  method  of  prime  or  ultimate  ratios,  or  by  the- ana- 
lytical method  of  derived  functions,  we  may  employ  infinitely  small  quantities  as  a 
sure  and  valuable  means  of  abridging  and  simplifying  our  demonstrations." 

*  The  student  will  be  much  benefited  by  plotting  curves  whose  equations  are 
of  the  form  y  =f(x),  and  interpreting  the  derivative  obtained  from  each  equation. 


DIFFERENTIALS. 


37 


PN=  sinaj ;  P'M=  sin  (x  +  dx) ;  therefore  P'S  =  d  sin  a;. 
0N=  cos  ic ;  0M=  cos  (ic  -|-  da;)  ;  therefore  NM=  —  d  cos  x. 
The  triangle  PP'^  is  a  right  triangle,  and  Z  PP'S  =  Z  POiV. 
Hence,  cZ  sin  a;  =  P'S  =  PP'  cos  PP'S  =  cos  a;  da;, 
and  d  cosa;  =  -  MN=-  PP'  sin  PP'^S 

=  —  sin  X  dx.         XII. 
AT  =  tana;, 
and  ^T'  =  tan  (x  -\-  dx) ; 

hence  TT'  =  d  tan  x, 

and  CT'  =  cZ  sec  x. 

BD  =  cot  X, 
and  JB^  =  cot  (x  -f  da;)  ; 

hence  HD  =  —  d  cot  a;, 

and  HE  =  —  d  cosec  a;. 

From  the  triangles  CTT'  and  ITDJ^;, 
similar  to  NOP,  the  differentials  of  the 
remaining  trigonometric  functions  may  be 
obtained. 

It  will  be  noticed  in  this  article,  that 
the  differential  of  a  function  is  negative  when  the  function  decreases 
as  the  independent  variable  increases. 


PROBLEMS. 

1.    If  the  side  of  an  equilateral  triangle  increases  uniformly  at  the 
rate  of  2  inches  per  second,  at  what  rate  does  the  altitude  increase  ? 

Let  a;  =  a  side  of  the  triangle,  and  y  its  altitude ;  then  2/^  =  f  x\ 
Differentiating,  and  solving  for  dy,  gives  dy  =  ■—-  dx,  which  shows  that 
if  an  infinitely  small  increment  is  given  to  x,  the  corresponding  incre- 
ment of  y  is  — -  times  as  great ;  that  is,  the  altitude  increases  ^~  times 
2i  2 


38  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 


as  fast  as  the  side.     When  x  is  increasing  at  the  rate  of  2  inches  per 

second,  y  is  increasing  at  the  rate  of  ^-  times  2,  or  V 3  inches  per 

second. 

# 

Remark.  In  these  examples  the  differentials  are  regarded  as  rates. 
The  rate  of  change  of  a  variable  at  a  given  instant  may  be  here  defined 
as  the  increment  which  it  would  receive  in  a  unit  of  time,  if  its  change 
should  be  uniform  throughout  the  interval.  Thus,  when  a  variable  at 
a  given  instant  is  said  to  change  at  the  rate  of  2  inches  per  second,  the 
meaning  is,  that  an  increment  of  2  inches  would  be  added  in  one  sec- 
ond, if  the  change  should  continue  uniform  for  one  second. 

2.  If  a  circular  plate  of  metal  is  expanded  by  heat,  how  rapidly 
does  the  area  increase,  when  the  radius  is  2  inches  long  and  increases 
at  the  rate  of  .01  inch  per  second  ? 

Let  X  =  radius,  and  y  =  area;  then  y  =  irx^,  and  dy  =  2trxdx. 
When  a;  =  2  inches,  and  dx  =  .01  inch  per  second,  dy  =  .04  tt  square 
inches  per  second,  which  is  the  rate  at  which  the  area  increases. 

3.  The  common  logarithm  of  1174  is  3.069668.  What  is  the  loga- 
rithm of  1174.8,  if  the  logarithm  is  assumed  to  change  uniformly  with 
the  number  ?  ^ 

Let  X  —  the  number,  and  y  =  its  logarithm ; 

then  y  =  log  x,  and  dy  =  ~  dx. 

Hence,  the  increment  of  the  logarithm  is  —  times  as  great  as  the 
increment  of  the  number. 

Therefore         dy  =  '^'^^^^^^  X  .8  =  .000295. . . . 

And      log  1174.8  =  3.069668  +  .000295  =  3.069963. 

Remark.     It  will  be  seen  from  the  equation  dy  =  —  dx,  that  as  the 

X 

number  increases  by  equal  constant  increments,  the  logarithm  will 
increase  more  and  more  slowly.  So  the  assumption  made  in  the  last 
example  is  not  strictly  true,  but  for  comparatively  small  changes  in  the 


O-  CI 


/^=   ^^-' 


..4  r--'     '-^-^     •^-r— ^—      -^      <L.  .'.     .^^Z)    '    -z: 


C.cj-t>  .    X    ^  A    ,    ' 


■vC*j^        f  cJT        c-j 


A,  m..—i   2Z~  a^  c     /  '  -  f( 


DIFFERENTIALS.  39 

number,  the  results  are  sufficiently  accurate  for  practical  applications. 
The  use  of  the  Tabular  Differences  in  tables  of  logarithms  is  based  on 
this  assumption. 

4.    In  the  parabola  y^  — 12  a?,  find  the  point  at  which  the  ordinate 

and  abscissa  are  increasing  equally.  Ans.  The  point  (3.6). 

■  5.    At  what  part  of  the  quadrant  does  the  arc  increase  twice  as 
rapidly  as  its  sine  ?  Ans.  At  60°. 

6.  The  logarithmic  sine  of  30°  5'  is  9.700062.  What  is  the  logar 
rithmic  sine  of  30°  6'  ?  Ans.  9.700280. 

7.  A  boy  is  running  on  a  horizontal  plane  directly  towards  the 
foot  of  a  tower  at  the  rate  of  5  miles  per  hour.  At  what  rate  is  he 
approaching  the  top  when  he  is  60  feet  from  the  base,  the'  tower  being 
80  feet  high  ?  Ans.  3  miles  per  hour. 

8.  A  vessel  is  sailing  northwest  at  the  rate  of  10  miles  per  hour. 
At  what  rate  is  she  making  north  latitude  ? 

Ans.  1.^1  -f-  miles  per  hour. 


/t 


^^   ^N 


^ 


/'■-i  ^"^ 


-        r      I  ,^ 


^  .  / 


'I 


-'      "^     -     y 


J 


<  CHAPTER   V. 

INTEGRATION. 
Art.  29.     Definition. 

Integration  is  the  operation  of  finding  the  function  from  which  a 
given  differential  has  been  obtained.  The  result  of  the  integration  is 
called  the  integral  of  the  differential.  The  symbol  which  indicates  the 
operation  of  integration  is  I  .  Since  differentiation  and  integration 
are  inverse  operations,  the  symbols  d  and  j ,  as  signs  of  operations, 
neutralize  each  other.* 

The  process  of  integration  is  of  a  tentative  nature,  depending  on 
a  previous  knowledge  of  differentiation;  just  as  division  in  arithme- 
tic is  a  tentative  process  depending  on  a  previous  knowledge  of 
multiplication. 

For  example,  d  {x'^)  =  4  y^dx ; 

therefore  (  4a^da;  =  cc^ 


/< 


Art.  30.     Fundamental  Formulas. 

The  fundamental  formulas  for  integration  are  obtained  directly 
from  the  formulas  for  differentiation.  A  function  is  the  integral  of 
a  differential,  if  the  function  when  differentiated  produces  the  differ- 
ential. All  integrations  must  ultimately  be  performed  by  the  formulas 
of  this  article.  When  a  differential  is  to  be  integrated,  if  it  is  not 
apparent  on  inspection  what  function  when  differentiated  produces  it, 


*  The  symbol  /  is  derived  from  the  initial  of  the  word  "summation."  Leibnitz 
introduced  the  letter  S  to  denote  the  operation,  and  this  gradually  became  elon- 
gated into  the  symbol  J". 

40 


«         VII. 
«       VIII.  a. 


INTEGRATION.  41 

the  differential  must  be  transformed  into  some  equivalent  expression 
of  known  form,  whose  integral  is  given  by  one  of  the  fundamental 
formulas. 

1.  C(du  H-  dv  —  dw)  =:u  +  v  —  w'j  from  I. 

2.  ladv  =  av;  "  II. 

3.  I  uav'''hlv  =  av"; 

4.  C^  =  a\ogv; 

J       V 

5.  r aMogadv  =  a";  **  IX.  a. 

6.  |Vdv  =  e%  «  IX.  6. 

7.  I  cosvd?;  =  sinv;  «  ^       XI. 

8.  i  —  sinvdv  =  cosv]  "  XII. 

9.  j  sec-vdv  =  tanv;  "  XIII. 

10.  I  —  cosec^vdv  =  cotv;  "  XIV. 

11.  j  sec  V  tan  v  dv  =  sec  iJ ;  "  XV. 

12.  j  —  cosec'ycotvdv  =  cosecu;  "  XVI. 

13.  j  sin  ?; d?;  =  vers  V ;  ^                                   "  XVII. 

14.  I  ~  cos V dv  =  coyevsv'j  "  XVIII. 

15.  f     ^^       =  arc  sin  ^;  "  XIX. 

16.  f ^^  =  arccos^;  «  XX. 


42 


17 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

from       XXI. 
XXII. 


arc  tan  v; 


rs.     I —^  =  arc  cot  V : 

J       1  -\-ir 

Y      dv 

^  vVv^  —  1 

r        dv 

•^      v\^v^  - 

f 


19 


20 


=  arc  sec  v ; 


arccosecv: 


V2v 


■/■ 


22.      I  - 


dv 


arc  vers 'y; 


=  arc  covers  v ; 


XXIII. 

XXIV. 

XXV. 

XXVI. 


^2v-v^ 

Art.  31.     Elementary  Rules  of  Integration. 

The  first  four  rules  of  integration  will  be  demonstrated  in  fulL 
(1)   By  I.,  d(u-\-v  —  w)  =  da  +  dv  —  dw  ; 

hence  I  d(u-\-v  —  w)  =  I  (du  +  dv  —  dtv), 

u  +  v  —  w=:  I  (du  +  dv  —  dw). 
u-{-v  —  w=  I  du-\-  I  dv—  i  dw, 
j  (du  +  iv  —  dw)  =  I  du-\-  I  dv—  I  dw. 


or 

But 
therefore 


Hence,  the  integral  of  the  algebraic  sum  of  any  number  of  differen- 
tials is  equal  to  the  algebraic  sum  of  their  integrals. 


(2)   BylL, 

d(av)  =  adv'j 

hence 

j  d(av)  =  j  adVf 

or 

av=  j  adv. 

But 

av  =  ajdv 

therefore 

j  adv  =  aj  dv. 

INTEGRATION.  43 

Hence,  the  integral  of  the  product  of  a  constant  and  a  differential 
is  equal  to  the  product  of  the  constant  and  the  integral  of  the 
differential.     . 


(3)   By  VIL, 

dav"  =  nav^'-Hv. 

Then 

1  dav"  =  1  nav'^'^dv  ; 

refore 

1  nav''-Hv  =  av'^. 

Hence,  when  a  function  consists  of  three  factors,  —  viz.  a  constant 
factor,  a  variable  factor  with  any  constant  exponent  except  —1,  and  a 
differential  factor  which  is  the  differential  of  the  variable  without  its 
exponent,  —  its  integral  is  the  product  of  the  constant  factor,  by  the 
variable  factor  with  its  exponent  increased  by  1,  divided  by  the  new 
exponent. 

(4)   ByVIII.a,  da\ogv  =  a—' 

Then  -         Cda\ogv=C—', 

therefore  | =  a  log  v. 

Hence,  the  integral  of  a  fraction  whose  numerator  is  the  product 
of  a  constant  by  the  differential  of  the  denominator,  is  equal  to  the 
product  of  the  constant  by  the  Naperian  logarithm  of  the  denominator. 

Art.  32.     Coj^stant  of  Integration. 

By  III.,  it  is  seen  that  the  differential  of  a  constant  is  zero  ;  hence, 
constant  terms  disappear  in  differentiation.  Therefore,  in  returning 
from  the  differential  to  the  integral,  some  constant  must  be  added, 
which  is  called  the  constant  of  integration.  The  value  of  this  arbitrary 
constant  is  determined  in  each  case  after  integration  by  the  data  of  the 
given  problem,  as  will  be  shown  hereafter.  So,  for  the  present,  the 
undetermined  constant  will  be  omitted,  but  its  addition  after  each  inte- 
gration will  always  be  understood.  Frequently,  when  a  differential  is 
integrated  by  different  methods,  the  results  may  not  appear  to  agree, 


44  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

but  on  inspection  it  will  always  be  found  that  the  integrals  differ  only 
by  some  constant. 

■♦ 

PROBLEMS. 
•* 

Formulas  1-3. 

1.     I  aMx. 


■/' 


U»^>A  By  Formula  3,  making  v  =  x,  and  n  =  4, 

'4 


<^     I'  ^''^'  CaMx  =  \C4.axHx  =  ^^' 


2.     Ch{Q  ax^  +  8  bx^)i  (2  aa;  +  4  bx^)  clx. 

diQax'-^S  bx^)  =  (12  ax  +  24  bx')dx', 

hence,  if  (2  ax  -f  Aba^)  dx  be  multiplied  by  6,  it  will  be  the  differential 
of  (6  aar^  4- 8  5a;^)^  without  the  parenthesis  exponent.  After  dividing 
the  constant  factor  by  6  to  preserve  the  same  value,  the  integration 
may  be  effected  by  Formula  3,  in  which  /v  =  6  ao?  -\-  8  b^x?. 

Therefore  Cb  (6  ax"  +  8  by?f  (2  ax  +  4  bx")  dx 

=  f-  (()  ar",  +  8  bx^) '  (12  ax  +  24  6a^  dx 

^(eax'-^Sbx^y^ 
=  - -g =  ^  (6  ax^ +86x^)1 

3 

3.  f    ^^^=  Ci(a'  +  x'')-hxdx=(a'-^a^i 

4.  C—^x~^dx.  Ans.  i^x'K 

5.  j  (^ax^  —  ^bx^)dx.  Ans.  ax^  —  bx^. 

Jdx 
—^'  Ans.  2-Vx. 


%.        a.)^      «*-*!         *-*  <^      ~^***       ^1*  ^^.~-**^aJ^^      fL^       X    o^"*^     <S. 


^  1*^         ,     ■■     .  ■      ■  .  ,        •      ■     ^     -       T  .*   ^^      ' 


Uo^.er''    '^  -^W.*«fc--*^**-«r**:^u.fiJi^       w .     c^      •(>**-*     .6;**-* 


9xx  = 


*****      i 


1 


0*-t-.0«*  A^>^  «:^  **-    -— »*  <>         *'**^' 


Tl^..-* 


/TT      <»i2-»«a.  «-  «-         ' "^ 


(«—        -<^    X       -*- 


/. 


^-;x^  .x'>'''o^''--'''^ 


^"-C  X     li'^^ 


t.- 


<>• 


•t  ( .. .   ^ 


-*.*^ 


W^    MU..        .,^«u.^      .^     ^  ^^    ^^^^     lu^    ^.^^ 


^     "ii^         <>l^*.'«A         _^^ t.a.» 


;  INTEGRATION.  45 

10.     C{Zax'  +  Uy?f^{2ax-\-^h7p)dx,       Ans.  -^ (3 aa^  +  4 6a^)i 
11      C         ^^^  Ans    _2aV36^T4^^ 


13. 


Formulas  4-6 

r  &Mx 


6  +  2a;3  .. 


14 


.     fJ^.  Ans.  log  (a; -a). 

*/  a?  —  a 


ft-i 


15.     r^l^.  ^ns.  —  log(a+6a^). 

^^      r\xdx^  Ans.lo^ix'  +  l)^. 

Jx^  +  i  * 

17.  r(loga;)3^.  Ans.  i(loga;)\ 

18.  ri^^.  J:„«.  log  (3  a;* +  7) a. 

19.  fi^.  -         ^«.s.  »-|  +  |-log(x  +  l). 

20.  /6a-d.  =  ^Ja-. loga.2cte  =  JJ-,  by  FormulaS.^^^^^   ^    ^ 

Se'^da;.  Ans.  3e*.                          "^ 

«2.     Cbe'^dx.  Ans.  -e*". 

J  a 


46  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

23.   jSa^'xloga'dx.  Ans.  fa< 

"24.     fa'^dx.  Ans.  __^^ 

J  1  4-  Inc 


l  +  loga 
Formulas  7-14. 


25.     1  cos  mxdx. 


/cos  mxdx  =  —  I  cos  mx-dmx  =  —  sin  ma;,  by  Formula  7. 
mj  m  '    '' 

26 .  fsin^  (2  a;)  cos  (2  a;)  (^a;. 

fsin^  (2  a;)  cos  (2  a;)  dx  =  ifsin^  (2a;)  cos  {2x)  2  da; 

=  I- fsin^  (2  a;)  d  sin  (2  a;)  =  |  sin*  (2  a;). 

27.  rsec2(a;3>)a^(^^  ^^^   J  tan  a;'. 

28.  j  5  sec  (3  a;)  tan  (3  a;)  (Za;.  ^ns.  |  sec  (3 a;). 

„o      Tsin  (3  a;)  da;  .        ,        ^ 

2^-     I 0,0   X  •  ^^«-  i sec 3a;. 

J    cos2(3a;)  ^ 

30.     1  e'=°**sina;c?a;.  ^ns.   —  e<=«''*. 


J(l  +  cos  a;)  da;  a  -i      r     ,     -      -^ 

- — ' 7-^ — •  Ans.  loff  ra;  +  sina;l. 
a;  +  sina;                                                        ^               ■■ 

I  tan  a;  da;.  Ans.  log  sec  a;. 


32.     I  tan  a;  da;. 


33.  j  sin  B  sec'^  6  dO.  Ans.  sec  0. 

34.  I  cot  a;  da;.  Ans.  log  sin  a;. 


n^      C  (l^         C  ^^  r^sec^(^ a;)  da;     ,       .      , 

35.     I-, —  =  1^-7-^^ r-=  I  ^    .    ^V —  =  logtanJa;. 

*/  sin  x     J  Z  sm  \  x  cos  \x    J       tan  \x 


36.     f-^  =  r     f      ^  =  log  tanf?  +  i»\ 


K.    ^'>'  J    6-  a 


<^K 


t,  K 


/^       ^C.^.        ^^K     S     -       -         ^C-'^V      .^         ---*      f^ 


3»7      fihr 


ii.V. 


*    X. 


**«&!^ 


f^         Jb.il-,vx, 


>  *jr. 


37 


INTEGRATION.  47 

.     I— Ans.  log  tan  a;. 

J  sm  a;  cos  a; 


38.     I -7-T — ^— :; —  Ans.  tana;  — cota;. 

J  SI] 


39.      ■  ^^ 


'*  J  V^ 


6V 


Formulas  14-22. 


•    6a; 
=  -  arc  sm  — 

a 


/da;         ^  r        g         _  1  r     a  ^  1 


In  order  to  integrate  the  preceding  differential  by  Formula  15,  it 
must  be  transformed  into  an  equivalent  differential  having  unity  for 
the  first  term  under  the  radical  sign,  and  having  for  its  numerator  the 
differential  of  the  square  root  of  the  second  term  under  the  radical 
sign. 


40.      f         ^^  . 

r      dx       _  r       a        _l  r      ^ 


^"^  ^dx 


1  hx 

=  -  arc  sec  — ,  by  Formula  19. 
a  a 


— Ans.  .-  arc  vers  — • 


42.  C_^l^_.  Ans.  arc  sin  (a;^ 

43.  r ^^  Ans.  arc  cos  (2V^). 

*^       ^x  —  ^y? 

44.  r^^^.  Ans.  ^arctan(ar^. 

45.  J      8a;  "^dx  _^  ^^^   4 V6 arc  vers  (6 a;*). 

^2a;7_6a;* 


48  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

2dx 


46.     (  — -•  Ans.  arc  cot  -• 

J       4  +  aH  2 

^'^-     I  r; — TT o'  -4*^''^-  arctan(a;  — 1). 

J  2  —  2x-{-  yr  ■  ^ 


dx  .         1  ex 


48.     I :z=z==z'  Ans.  -!^arcsec 


Vc-x2  _  ^2^2  ah  ah 

r dx_ 

^       Va^x  — 


Aa       I  (^^  .1  2h^x 

49.     I •'  Ans.  -arc covers — —  • 


"6V  h 

2 
V3* 


50.     I =  I ; — -  =  — =arc  tan  (x  +  I) 


Art.  33.     Integration  of  Trigonometric  Differentials. 

Trigonometric  differentials,  to  which  the  previous  formulas  cannot 
be  made  applicable  by  algebraic  reductions,  may  often  be  brought 
to  known  forms  by  trigonometric  reductions. 

Every  function  may  be  differentiated  by  a  general  method,  but 
there  is  no  general  method  of  integration.  Thus,  while  every  function 
may  be  differentiated,  but  a  limited  number  of  differentials  can  be 
integrated.  In  attempting  to  integrate  any  given  differential,  the 
object  is  always  to  transform  it  to  a  fundamental  form  whose  integral 
is  known. 

Hence,  the  processes  of  the  Integral  Calculus  are  transformations 
to  effect  reductions  to  fundamental  formulas.  In  order  to  become  pro- 
ficient in  these  operations,  it  is  necessary  to  have  much  practice  in  the 
solution  of  problems. 

PROBLEMS. 

1.  I  eos'xdx. 

lcos^xdx=  I  (^  -^  ^cos2x)dx  =  ^x-\-  \sm2x. 

2.  j  sin^a;(?a;.  Ans.  Jic  — Jsin2a;. 

3 .  I  tan'^  X  dx. 

itsai^xdx=  j  (sec^a;  — l)tana;da;  =  ^tan'^a;  — logsecic. 


INTEGRATION.  49 

4.  itsin^xdx.  Ans.  tana;  — aj. 

5 .  I  tan*  X  dx.  Ans.  ^  tan^  x  —  tan  a;  +  a;. 

6 .  I  tan^  x  dx.  Ans.  \  tan*  x  —  ^  tan^  x  +  log  sec  x. 

7.  I  sin^xdx. 

isiu^xdx=  I  (l  —  cos^xysmxdx=  j  (— 1  +  2cos^a;  — cos'*x)dcosa; 

,   9       o        cos'' a; 

=  —  cos  a;  +  I  cos^  x 

5 

'  x  dx.  Ans.  sin  x  —  ^  sin^  x. 


10 


.      I  COS^; 

.     j  cos^  X  dx.  Ans.  sin  x  —  sin^ «  +  f  sin^  ^  —  t  sin'  x. 

.     j  cot*  a;  da?.  Ans.   —  Jcot^a;4- cota;  +  a;. 

11.     rcos*a;da;=  ("(^  +  icos2a;)2(^a;  =  Ja;  +  isin2»4- i  fcos^ (2 a;)c? (2a;) 
=  J  a;  4-  ^  sin  2  a;  -f-  1^  [a;  +  ^  sin  4  a;]  =  I  a;  +  ^  sin  2  a;  +  -5^2  sill  4  a;. 


Art.  34.     Definite  Integrals. 

It  was  shown  in  Art.  32,  that  an  arbitrary  constant  must  be  added 
after  each  integration.  Before  the  value  of  this  constant  is  determined, 
the  integral  is  said  to  be  indefinite. 

If,  from  the  data  of  the  given  problem,  the  value  of  the  integral  is 
known  for  some  particular  value  of  the  variable,  the  constant  can  be 
determined  by  substituting  this  value  of  the  variable  in  the  indefinite 
integral. 

For  example,  let 

ds 

—  =  gt  -\-v',  in  which  g  and  v'  are  constants. 


50  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

By  integrating  and  adding  the  constant  C, 

*'   Now  if        S  =  S'  when  t  =  0,  C  will  be  equal  to  /S', 
and*  S  =  igt^-{-v't-{-S'. 

If,  in  any  indefinite  integral,  two  different  values  of  the  variable  be 
substituted  for  the  variable,  and  the  result  given  by  the  second  substi- 
tution be  subtracted  from  the  first  result,  the  constant  of  integration  is 
eliminated,  and  the  integral  is  said  to  be  taken  between  limits. 

The  definite  integral  of  f'(x)dx  between  the  limits  a  and  b  is  indi- 
cated thus : 

'^y(x)dx,  (1) 


JT 


in  which  a  is  the  superior  limit  and  b  the  inferior  limit  of  integration. 
In  (1),  /'(ic)  dx  is  first  to  be  integrated,  then  a  and  b  are  to  be  succes- 
sively substituted  for  x,  and  the  second  result  is  to  be  subtracted  from 
the  first. 

It  is  assumed  that  the  integral  is  continuous  between  the  limits  a 
and  b.  A  function  is  said  to  be  continuous  between  two  values  of  the 
variable  when  it  has  a  single  finite  value  for  every  value  of  the  varia- 
ble between  the  given  values,  and  changes  gradually  as  the  variable 
passes  from  the  first  value  to  the  second.  Evidently,  the  value  of  the 
integral  up  to  the  superior  limit  includes  the  value  of  the  integral  at 
the  inferior  limit.  Hence,  the  difference  between  the  values  of  the 
integral  at  two  limits  will  be  the  value  of  the  integral  between  those 
limits. 

Assuming  that  the  inferior  limit  is  equal  to  b,  and  writing  the  inte- 
gral in  the  two  different  ways ; 

ff'{x)dx=f(x)  +  C,  (2) 

and  jy'(x)dx  =f(x)  -  [/(x)],  (3) 

If  these  two  forms  are  taken  to  represent  the  same  quantity, 

f(x)  +  C  =  f(x)  -  [/(a;)],;  whence  C=  [/(x)],.  (4) 


INTEGRATION. 


51 


Thus  it  will  be  seen  that  the  upper  limit  is  any  final  value  of  the 
increasing  variable  x,  and  that  the  lower  limit  may  be  assigned  without 
defining  the  upper  limit.  Equation  (4)  shows  that  the  constant  C  de- 
pends on  the  lower  limit.  Therefore,  the  integral  in  equation  (2)  Is 
indefinite  because  a  free  choice  is  left  with  regard  to  the  selection  of 
both  limits.  The  part  f(x)  depends  on  the  value  of  x  selected  for  the 
superior  limit,  and  the  part  C  depends  on  the  value  taken  for  the  infe- 
rior limit. 


Art.  35»     Geometric  Illustration  of  Definite  Integration. 

The  problem  of  finding  the  areas  of  plane  curves  was  one  of  those 
that  gave  rise  to  the  Integral  Calculus,  and  this  problem  furnishes  an 
illustration  of  the  preceding  article. 

In  Fig.  7,  let  MN  represent  any  plane  curve ;  it  is  required  to  find 
the  area  included  between  the  curve,  the  X-axis  and  two  ordinates. 

Let  (aj,  y)  be  the  coordinates  of  the  point  F.  If  BS  =  Aa  be  added 
to  Xf  SQ  =  y  +  Ay.     QC  and  FD  are  drawn  parallel  to  the  X-axis. 

If  A  represents  the  area  of  the  curve  between  two  ordinates  and  the 
X-axis,  AA  =  area  BFQS. 
Y1 


C_0_- 


R     S 


W 


Fig.  7. 


Then 


BCQS  ^CR^y-hAy^.      Ay 
BFDS     FR  y  y' 


Now,  as  Ax  approaches  zero,  Ay  also  approaches  zero  j 

and  limit  ^^^  =  1. 

FFDS 


62  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

But  the  area  RPQS  is  intermediate  between  area  MCQS  and  area 
RPDS',  hence 

RPDS       ' 


or'  limit -A4_  =  i 

y '  ^x 

dA      . 
or  -  —r  =  ^'^ 

yax 

therefore  dA  =  ydx, 


and 


A==Jydx-ha  (1) 

If  the  area  between  the  ordinates  NW  and  ML  is  required,  the 

superior  limit  will  be  the  abscissa  0  W,  and  the  inferior  limit  will  be 

the  abscissa  OL.     If  these  limits  are  respectively  a  and  6,  the  area  will 

be  denoted  by  * 

,.  A=JJydx.  (2y 

Let  the  particular  curve  whose  area  is  required  be  the  common 
parabola,  then  y  =:  ^2px.     Substituting  this  value  of  y  in  (1),  gives 

A=  CV2pxdx+0  =  V2p  Cx^dx-{-  (7  =  |V2pa;t  +  0. 

If  the  area  is  estimated  from  the  origin,  when  a;  =  0,  A  =  0;  hence, 
by  the  first  method  of  Art.  34,  (7=0,  and,  therefore,  A  =  | V2  j9a;^. 

If  the  area  is  required  between  two  ordinates  whose  abscissas  are  a 
and  6,  by  the  second  method  of  Art.  34, 

j^^/2^dx  =  [I  V2^a;f]'  =  |V2^  [a^  -  6^]. 

Art.  36.     Change  of  Limits. 
Let  Jf\x)dx=f{x); 

then  JJr(x)dx=f(a)--f(b)  ;.:*.;.' 

■ ■-       ■ [/(&)-/(«)]■:;:    '.:^  .-: 

=^-£f'(x)dx.  - 


.     .    ^.    .     INTEGRATION.  53 

Hence,  the  limits  of  integration  may  be  interchanged  by  changing  the 
sign  of  the  integral. 

It  may  also  be  readily  shown  that 

j'  f  (X)  dx  =£  r  ix)  dx  +  jT'  /'  {X)  dX. 

If  a  new  variable  be  substituted  for  the  old  variable  in  integration 
between  limits,  corresponding  changes  must  be  made  in  the  limits  of 
integration. 

For  example,   I    x'^dx  is  required. 

Suppose  x  =  z^]  then  when  x  =  4,  2;  =  ±  2, 
and  when  x  =  1,  z  =  ±  1, 


Therefore  f  x'^dx  =  2  f^  z^^'+^dz 


n  +  l 


PROBLEMS. 


1.  Find  the  particular  integral  of  dy=(x^—b^x)dx,  ii  y=0  when  x=2, 

Ans.  y  =  t-^^2b'-4, 
4        2 

2.  Find  the  particular  integral  of  du = (1  -|- 1  ax) ^dx,  if  ?^ = 0  when  a;= 0. 

I  ».  jr(.--,...[=p-g;=l'. 

2  r''l2  -4:e)de  =  0.64.  8.     f^^^M  =  V2  - 1. 


i       4 


.     r6T^dx  =  3S.  9.     p:^da;  =  4 

J2  Jo      Va; 

*/o    a^  +  a^     2  a  'Jo  ^^2  _  ^a, 


—  \rT, 


f  J2  l+ar"        2  Jo  " 


54  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

—  .  assume  z—--  Ans.  f . 

13.^  In   I     sinxcos^ajdic,  assume  siiia;  =  2!.  Ans.  \. 

14.  In   C    ^^^     .  assume  2/  =  l-ar'.  ^ns.   ±  1. 

15.  Find  the   area  of  the  curve  y  —  Q^-\-^Xy  between  the  abscissas 

x=^^  and  a;  =  0.  -4ns.  180. 

Note.  Applications  of  the  Integral  Calculus  in  rectifying  curves,  determining 
areas,  volumes,  centre  of  mass,  and  moment  of  inertia,  will  be  found  in  Chapters 
XIX.,  XX.  and  XXI. 


CHAPTER  VI. 
SUCCESSIVE  DIFFERENTIATION  AND  INTEGRATION. 

Art.  37.     Successive  Derivatives. 

As  the  derivative  of  a  function  is,  in  general,  a  new  function  of 
the  independent  variable,  it  can  be  differentiated.  The  derivative  of 
the  first  derivative  is  called  the  second  derivative.  Likewise,  when  the 
second  derivative  is  a  function  of  the  independent  variable,  it  may 
also  be  differentiated,  giving  the  third  derivative;  ana  so  on. 

For  example,  if  y  =  ax*; 

•V. 

dx 

dx  \dxj 


^rd^fdf 

dx\_dx\clx^ 


=  24  ax,  etc. 


The  symbols  for  the  successive  derivatives  are  usually  abbreviated 

as  follows : 

d^fdy\^^ 
dx\dxj     doc^^ 

dx\_dx\dxjj     dx\d7?J     da^ 

d  /d"-VA  ^  d"y 
dx\dx''-'^)     dx"" 

The  successive  derivatives  are  often  called  successive  differential 
coefficients.  As  the  first  derivative  is  often  denoted  by/'(ic),  the  suc- 
cessive derivatives  are  often  denoted  by/"(a;),/'"(ic),  etc. 

55 


56  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

If  differentials  are  employed,  successive  differentials  will  follow 
instead  of  successive  derivatives.  The  differential  obtained  immedi- 
ately from  the  given  function  is  the  first  differential ;  the  differential 
of  the  first  differential  is  the  second  differential ;  and  so  on.  If  the 
function  be  represented  by  ?/?  the  successive  differentials  will  be 
denoted  by  dy,  d-y,  d^y,  etc. 

In  successive  differentiation  it  is  customary  to  make  the  assumption 
that  the  differential  of  the  independent  variable  is  constant ;  i.e.  the 
independent  variable  increases  by  equal  increments,  and  hence  is 
called  an  equicrescent  variable.  The  independent  variable  will  always 
be  understood  to  be  equicrescent  unless  the  contrary  is  explicitly 
stated.  ,  . 

Art.  38.     Successive  Integration. 

Two,  three  or  n  integrations  must  be  performed  in  order  to  obtain 
the  original  function  from  which  a  second,  third,  or,  in  general,  an  nth 
derivative  was  derived. 

For  example,  if  ^  =  24  ax.  (1) 

Integrating  (1),  ^  =  12a3^:      "  _  (2) 

Integrating  (2),  ^  =  4aa^.  ....    .   ,      .,,  .     -  (3) 

Integrating  (3),  y  —  ax^.  (4) 

But  an  arbitrary  constant  should  be  added  after  each  integration, 
as  a  constant  term  may  have  disappeared  at  each  differentiation. 

Then,  in  general,  let  — ^=/(a;);  and  denote  the  successive  inte- 
grals of  the  function  by  /i(a;),  /2(a?),  /sCa;),  etc.,  and  the  constants  of 
integration  by  Cj,  C2,  C3,  etc. 

«^-°  S=-^(^)^       ■ 

then  g^=/,(a;)+Q, 


SUCCESSIVE  DIFFERENTIATION  AND  INTEGRATION.       57 


d--'y_ 
dx--' 

--Ai^)  +  C,x  +  02, 

d--''y_ 
dx--^ 

=fz{^)  +  C,'^+C^+C,, 

and 

finally, 

Q(f 

1-1 

.r"-2 

I.2.3.. 

.(n-1)  '   ^^1.2 

3...(n- 

-2) 

PROBLEMS. 

1. 

y  =  aaf. 

2.      2/  = 

=  tan  X. 

^^nax^' 
dx 

1. 

> 

dx 

=  sec^a;; 

%=<^- 

-1) 

aa;"-2; 

dx^ 

=  2  sec^  X 

•  tana;; 

+  C7„. 


^^  =  n(n  -  l)(n  -  2)  aaf»-« ;  ^3,, 

•  ••  •••  ,_.  .»=" 

^  =  k.a.  ^  =  8tana;sec2a;(3sec2a;-l). 

da;"     ■-  dx^  ^ 

3.  y  =  ax^  +  ha?.         '     "  —  " .      ' 

4.  ?/ =  log  (x  +  1).  • 

5.  2/  =  a*.  ;       -  .        : 

6.  ?/  =  6a;*  — 4iB*  — 6aj*. 

8.  ?/ =  a;^  log  (aj^. 

9.  2/  =  tan^a;  +  81ogcosa;4-3a^. 


:0. 

d'y  _ 
dx* 

:-6(a;4-l)-^ 

d-y^ 
dx- 

=  a%togay. 

d'y  _ 
dx'' 

=  144. 

^y_ 

24a;(l-ar^ 

d^ 

(1+aO* 

d^y 
da^ 

_48^ 
a; 

d'y 
dx' 

=  6  tan*  a;. 

68  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

10.  y  =  xe".  i!y=:(x-\-n)  e\ 

11.  ?/  =  e*"°''«cos(a;siiia).  ^  =  €== '=<^'"»  cos  (a;  sin  a  +  a) ; 
,  dx 

^  =  e '"« « cos  (a;  sin  a  +  na). 

13.  ^  =  2x-8.  2/  =  logx+C,^'  +  C2X  +  C3. 

14.  ^=cosa;.  2/  =  cosx  +  Oi-  +  C,^  +  C.x  +  04. 

fl_^L_^Ur  =  0.     16.  ^ =  .6366  a,  when  2/  =  a  sin  a;. 

17.   Find  value  of  t  from  a^i^  =  &  (c  -  a;). 

de        ^         ^ 


Integrating  once  gives 

'dt^- 


a^=6(2ca;-ar^; 


hence  ^^  =  V?        ^"^ 


Therefore  t  =  -v/^arc  vers  -• 

6  c 


-4' 


18.   In  the  harmonic  curve  whose  equation  is  ^  =  rj  sin  mZ  +  ^s  cos  m?, 

find —-;  ri,  rg  and  m  being  constants.  ,„, 

^^  Ans,  ^  =  -m%. 


^  ^^f' 


APPLICATIONS  IN   MECHANICS.  69 

APPLICATIONS  IN  MECHANICS. 
Art.  39.     Velocity  and  Acceleration  of  Motion. 

The  mean  velocity  of  a  moving  body  for  a  certain  period,  is  equal  to 
the  distance  passed  over  expressed  in  some  unit  of  length,  divided  by 
the  length  of  the  period  expressed  in  some  unit  of  time.  The  velocity 
is  uniform  if  equal  distances  are  traversed  in  equal  times;  and  the 
velocity  is  variable  if  unequal  distances  are  traversed  in  equal  times. 

Let  s  =  distance,  v  =  velocity,  and  t  =  time. 

And  let  As  denote  the  increment  of  distance  passed  over  by  the 
body  in  the  increment  of  time  A^,  while  the  velocity  has  increased  to 
v  -h  Av. 

The  distance  actually  passed  over,  if  the  velocity  is  variable,  lies 
between  the  distances  it  would  have  passed  over  if  its  velocities  at  the 
beginning  and  end  of  the  period  had  been  uniform  j  hence 

vAt<As<  (v  +  Av)At, 
and  v<—<{v-\-  Av). 

Now,  as  A^  approaches  zero,  Av  approaches  zero,  (v  +  Av)  approaches 

V,  and  —  approaches  — ;  and  as  the  middle  term  is  intermediate  be- 
Ac  (it 

tween  the  first  and  third  terms,  at  the  limit 

The  acceleration  at  any  instant  is  the  rate  at  which  the  velocity  is 
changing  at  that  instant,  and  since  the  derivative  of  a  function  meas- 
ures the  rate  at  which  its  value  is  changing,  if  the  acceleration  is 
denoted  by  a, 

dv  .n\ 

therefore  a  =  ^f^\  =  '^.  (3) 

dt\dtj    ae  ^  ' 

Equations  (1),  (2)  and  (3)  are  fundamental  formulas. 


60  diffe;rential  and  integral  calculus. 

Art.  40.     Uniformly  Accelerated  Motion. 

*  Motion  IS  uniformly  accelerated  when  the  acceleration  is  constant. 
Denoting  the  acceleration,  which  may  be  positive  or  negative,  by  g  j 
then  from  (3),  Art.  39, 

therefore  p  =  gt-^Ci.  (4) 

etc 

Cds=  Cgtdt  +  Cidtj 
therefore  s  =  yf-{-  Cit  +  Cj.  (5) 

If,  in  (4),  zero  be  substituted  for  t,  Ci  will  be  equal  to  (  —  ]      ;  or 

\dtJt=o 

Ci  will  be  the  velocity  at  the  beginning  of  the  period,  which  may  be 
represented  by  Vq. 

If,  in  (5),  zero  be  substituted  for  t,  C2  will  be  equal  to  the  left 
member ;  or  C2  will  be  the  distance  already  passed  over  at  the  begin- 
ning of  the  period,  which  may  be  denoted  by  Sy. 

Making  these  substitutions  in  (4)  and  (5), 

v  =  gt  +  Vo,  .  ^     (6) 

and  5  =  i  gt'^  +  v^,  +  So-  (7) 

PROBLEMS. 

1.  If  a  body  is  dropped,  what  distance  will  it  fall  in  5  seconds,  and 
what  will  be  its  velocity  at  the  end  of  the  fifth  second,  the  acceleration 
of  gravity  being  32.2  feet  per  second  ? 

In  (6)  and  (7),  Vq  =  0  and  So  =  0 ;  hence 

v^gt, 
and  s  =  \gt^. 

Substituting  the  values  of  g  and  ^,  gives 

V  —  161  feet  per  second,  and  s  =  402.5  feet. 

2.  If  a  body  is  projected  vertically  upwards,  to  what  height  will  it 
rise,  and  what  will  be  the  time  of  ascent  ? 


r  V 


APPLICATIONS  IN  MECHANICS.  61 

In  this  case,  the  acceleration  is  negative,  and  Sq  =  0 ;  hence  equa- 
tions (6)  and  (7)  become 

v  =  -gt-\-VQ,  (8) 

and  s  =  —  ^gf-{-v^.  (9) 

When  the  body  attains  its  greatest  altitude,  its  velocity  becomes 
zero.     Therefore,  if  v  =  0  in  (8), 

«  =  !»,  (10) 

which  is  the  time  during  which  the  body  rises. 
Substituting  t  =  ^  ux  (9),  gives 

.     .,.       ,         «  =  2T  W 

which  is  the  height  to  which  the  body  will  rise. 

3.  A  man  is  ascending  in  a  balloon  with  a  uniform  velocity  of  20 
feet  per  second,  when  he  drops  a  stone  which  reaches  the  ground  in 
4  seconds ;  find  the  height  of  the  balloon.  Ans.  176  feet. 

4.  A  body  is  projected  upwards  with  a  velocity  of  80  feet  per 
second ;  in  what  time  will  it  return  to  the  place  of  starting  ? 

^^)  =:•  --   ;        . —  ::■■'  Ans.  5  seconds. 

5.  Two  balls  are  dropped  from  a  balloon,  one  of  them  3  seconds 
before  the  other;  how  far  will  they  be  apart  5  seconds  after  the  first 
one  was  dropped  ?  Ans.  336  feet. 

6.  A  body  when  first  observed  was  falling  at  the  rate  of  40  feet  per 
second,  and  struck  the  earth  in  5  seconds ;  required  the  entire  distance 
that  the  body  fell. 

Art.  41.     Derivatives  of  the  Product  of  Two  Functions. 

Let  2/  =  uv,  (1) 

u  and  V  being  functions  of  a;;  then,  by  IV.,  '     .-■-.- 

dx        dx         dx 


62  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Differentiating  (2)  with  respect  to  ic,  gives 

»^  ^_    ^  .dvdududv        d^v 

doc^        da^     dx  dx      dx  dx        dx^ 


dx^        dx  dx        dx^ 


Similarly, 


d^  _    d^  ,  dv  dhi  .2dv^,2—  —  j-—^'^M^'^ 
do?        dx^     dx  dx^        dx  dx^        dxF  dx     do?  dx     do? 

—    d?u     r^dA)^  d^u  .  c^dH  du     d^v 
da?        dx  do?        do?  dx     do? 


By  proceeding  as  above,  the  fourth  derivative,  and  other  successive 
derivatives,  may  be  obtained,  and  it  will  be  seen  that  the  same  law  of 
the  terms  applies,  the  numerical  coefficients  being  those  of  the  Bino- 
mial Theorem ;  giving  the  general  form, 

d"?/        d'^u  ,      dv  d'^'^u  .  n(n  —  1)  d?v  d*^~^u  , 
dof"        dx^        dxdx^-^  1-2      dx^  dx''-^ 

^nP^^^  +  pu.  (1) 

dx^'-^dx     dx""  ^^ 

This  is  known  as  Leibnitz's  Formula. 


That  (1)  is  true  for  any  nth  derivative  may  be  readily  proved  by  mathematical 
induction. 

Differentiating  (1),  and  arranging  the  terms,  gives 

dx»+^        dx^+^      ^  ^  dxdx^  1-2      dx'^dx^-'^      " 

If  the  law  of  the  terms  expressed  by  (1)  is  true  for  n,  it  then  appears  from  (2) 
that  the  formula  is  true  when  n  is  changed  into  n  +  1.  But  (1)  has  been  shown  to 
be  true  when  n  is  1,  2  or  3,  then  the  formula  must  be  true  when  n  is  4,  6,  6,  or  any 
positive  integer. 


APPLICATIONS  IN  MECHANICS.  63 

PROBLEMS. 

Find  the  derivatives   in  the  following  examples,  by  the  aid  of 
Leibnitz's  Theorem : 

2.   y  =  ^z.  p^e-fa-z  +  na'-^^  +  '^^^a'-'^...). 

dx"  V  daj         1 . 2  dx^     J 

Z.  y=^o?a',  ^  =  «*  (log  a)""'  [(aJ  log  a  +  nf-n]. 


^i 


CHAPTER   VII. 

FUNCTIONS  OF  TWO  OR  MORE  VARIABLES.     IMPLICIT  FUNCTIONS. 
CHANGE  OF  THE  INDEPENDENT  VARIABLE. 

Art.  42.     Partial  Differentiation. 

If  2  be  a  function  of  two  independent  variables  x  and  y,  it  may  be 
expressed  thus : 

z=f{x,y).  (1) 

In  (1),  z  may  be  changed  by  changing  either  x  or  y. 
Por  example,  in  the  equation  of  a  plane, 

z  z=  ax  -}- by  -\- Cf  (2) 

X  and  y  are  two  independent  variables,  of  which  2!  is  a  function.  In 
(2)  a  value  may  be  given  to  either  coordinate  x  ot  y  without  any  refer- 
ence to  the  other ;  so  if  either  x  or  y  receives  an  increment,  z  will  take 
a  corresponding  increment.  Then  z  may  be  differentiated  with  respect 
to  x  and  y  separately. 

If  (2)  be  differentiated,  supposing  x  to  vary  and  y  to  remain  con- 
stant, the  derivative  is  written 

!=«•  (^> 

If  (2)  be  differentiated,  supposing  y  to  vary  and  x  to  remain  con- 
stant, the  derivative  is  written 

|^  =  ?>.  (4) 

dy 

These  derivatives  are  called  partial  derivatives. 
According  to  the  differential  notation,  equations  (3)  and  (4)  may  be 
transformed  into 

-^dx  =  a  dx,    and     -^dy  -=  bdy, 
ax  dy 

and  these  expressions  are  called  partial  differentials. 

64 


l'    ^    2 ' 


FUNCTIONS  OF   TWO   OR  MORE  VARIABLES.  65 

Therefore;  a  partial  differential  of  a  function  of  several  variables  is 
a  differential  obtained  on  the  hypothesis  that  only  one  of  the  variables 
changes. 

A  total  differential  of  a  function  of  several  variables  is  a  differential 
obtained  on  the  hypothesis  that  all  of  the  variables  change. 

To  distinguish  between  the  partial  differentials  of  a  function,  the 

following  notation  is  adopted :  —  Ax  and  —  Ay  will  represent  partial 

dz  dz      ^^  ^y 

increments,  and  ^-dx  and  —dy  partial  differentials  of  z,  with  respect 
dx  dy 

to  x  and  y,  respectively. 

DxZ  and  [  —]  have  been  used  to  represent  total  derivatives  of  z  with  respect- 
to  X. 

The  general  equation  of  a  surface  as  given  in  Analytical  Geometry 
of  three  dimensions  is 

in  which  x  and  y  are  independent  variables. 

If  the  surface  be  cut  by  a  plane  parallel  to  the  XZ  plane,  the  equa- 
tion of  the  curve  of  intersection  will  contain  the  variables  x  and  z  only,- 

and  the  slope  of  the  curve  will  be  expressed  by  — 

dx 

Likewise,  the  equation  of  a  section  of  the  surface  parallel  to  the  YZ 

plane  will  contain  the  variables  y  and  z  only,  and  its  slope  will  be  — 

dy 

Art.  43.     Total  Differential  of  a  Function  of  Two 
OR  More  Independent  Variables. 

Let  z=f(x,y). 

Let  X  and  y  be  given  successive  increments  Ax  and  Ay,  and  repre- 
sent the  corresponding  total  increment  of  the  function  by  Az, 

Let  z'=f(x-^AXjy)', 

then  ^  Ax  =f(x  +  Ax,  y)  -fix,  y),  (1) 

Az' 

^  Ay  =f{x  -{-Ax,y-\-  Ay)  -f{x  +  Ax,  y),  (2) 

Ay 


H  ** 


66  DIFFERENTIAL   AND  INTEGRxiL   CALCULUS. 

and  A2!  =f{x  +  Ax,  y -\-  Ay)  -  f(x,  y).  (3) 

Adding  (1)  and  (2),  and  placing  the  first  member  of  "the  resultant 
equation  equal  to  the  first  member  of  (3),  gives 

Ax  Ay    ^  ^  ^ 

Now,  if  Ax  and  Ay  approach  zero,  limit  Az'  —  limit  Az, 

therefore,  dz  =  —-dx-\-  ^-cly. 

ox  dy 

Hence,  the  total  differential  of  a  function  of  two  variables  is  equal 
to  the  sum  of  its  partial  differentials. 

Similarly,  the  total  differential  of  a  function  of  any  number  of 
independent  variables  may  be  found  to  be  equal  to  the  sum  of  its 
partial  differentials. 

PROBLEMS. 

1.  zz^aary^. 

^dx=^3axFy^dx:  —dy  =  2ax^ydy. 
dx  dy 

therefore  dz  =  S  aa^y^dx  +  2  ax^y  dy ; 

2.  z  —  T?.  dz  =  yx^-^dx-\-af\ogxdy. 

3.  ^  =  arctan?.  <i^  =  E%lll^. 

X  XT  -\-y^ 

4.  z  =  sin  (osy)>  dz  =  cos  (xy)  [^ydx-\-  x dy"]. 

5.  2=3  2/"^*.  dz  =:y'''"' logy  cos  xdx-\-^^^^dy. 

if 

6.  M  =  x".  du=  x^'~^(yzdx 4- zx logxdy -}- xylogxdz). 

Art.  44.     Total  Derivative  of  u  with  respect  to  x  when 

^  =/(a?,  y,  2),  y  =  <f>  (x),  and  z  =  0i(aj)." 

By  Art.  43, 

du  =  ^dx  +  ^-^dy-\-^dz', 
dx  dy  dz 

thpfp  dM^du.dudy^.du  dz^  « >. 

dx      dx      dy  dx      dz  dx 


FUNCTIONS  OF   TWO   OR  MORE   VARIABLES.  67 

Cor.  1.   If  u  =f(x,  2/),  and  y  =  <f>(x), 

therefore  •  ^^Su^dudy_ 

ax      ox      ay  ax  ^  ^ 

Cor.  2.   If  u=f{y,  z),    y  =  <f>(x),  and  2  =  <^(a;), 

dy  dz  .. 

therefore  du^dud^_^eu^_  , 

dx      dy  dx      dz  dx  - 

Cor.  3.  If  u  =/(2/),  and  y=<f>  (x), 

du  =  —  dy: 
dy   ^' 


therefore  du^dudy^ 

dx     dy  dx 


(4) 


In  the  proposition,  u  is  directly  a  function  of  x  and  also  indirectly 
a  function  of  x  through  y  and  z. 

In  Cor.  1,  u  is  directly  a  function  of  x  and  indirectly  a  function  of 
X  through  y. 

In  Cor.  2,  u  is  indirectly  a  function  of  x  through  y  and  z. 

In  Cor.  3,  u  is  indirectly  a  function  of  x  through  y. 

PROBLEMS. 

1.   u  =  e'"(y  —  z),  y  =  a  sina;,  and  z  =  cos x, 

du  _du      du  dp      du  dz 
dx      dx      dy  dx      dz  dx 

du  ax/  ^    du        _-    du  „ 

dy  dz 

.   dx  dx 


68  DIFFERENTIAL   AND   INTEGRAL  CALCULUS.       . 

therefore  —  =  ae"'  (y  —  z)  -\-  ae"^  cos  x  -\-  e"""  sin  x 

dx 

^^  =  e''^{a^  sinx  —  acosa;  +  acosa;  +  sina:) 

=  e"''  (a^  +  1)  sin  x. 

2.  w  =  arctan(a;i/),  and  y  =  e*  du  ^  e' (1 -\- x) ^ 

3.  u  =  yz,  y  =  e%  and  z=x'^-^x^-\-12^-2^x+14..     ^  =  eV. 

da; 

4.  w  =  log  (r^  -  2/2)^  and  2/  =  rsin^.  ^  =  -2tan^. 

dv 

5.  w  =  — )f-——l     2/  =  a  sin  a;,  and  2f  =  cos  a;.  —  =  e**  sin  a;. 

a^  H-  1  da; 


Art.  45.     Successive  Partial  Derivatives  of  Two  or  More 

Variables. 

If  u  =f(Xy  y\  then  —  and  —  are,  in  general,  functions  of  both  x 
ox  oy 

and  ?/,  and  may  be  differentiated  with  respect  to  either  independent 
variable,  giving  second  partial  derivatives. 

The  partial  derivative  of  —  with  respect  to  x  is  — (—]z=—. 

dx  dx\dxj       dse^ 

The  partial  derivative  of  —  with  respect  to  w  is  — ( —  )  =  — • 

dy  dy\dyj      5/ 

The  partial  derivative  of  —  with  respect  to  ?/  is  — f  —  ]  =  — —. 

dx  6y\dxJ      dydx 


■  The  partial  derivative  of  —  with  respect  to  x  is  — ( —]  =  — —- 

ay  dx\dyj      dxdy 

Likewise,  ^  is  a  third  partial  derivative,  obtained  by  three  suc- 
cessive differentiations ;  first,  with  respect  to  x  regarding  y  as  constant, 
and  then  twice  with  respect  to  y  regarding  x  as  constant. 

dy\dxdyj      dydxdy     dx\dxdy^J      dx^dy^^ 
and  similarly  with  all  other  partial  derivatives. 


IMPLICIT  FUNCTIONS.  69 

Art.  46.     If  u=f(x,y).  to  prove  that  ^    ^    =  ^    ^  » 

On  the  supposition  that  x  alone  changes  in/(a;,  y), 

^u  ^f(x  +  Ax,  y)  -f{x,  y) 
A.^  Ao; 

Now,  supposing  y  alone  to  change  in  (1), 

_A_ /Aw\ ^/(a;4-Aa;,  y-}-Ay)-f(x,  y-{-Ay)-f{x-\-^y)-{-f(x,  y) 
Ay\^xj  Ai/  •  Aa; 

On  the  supposition  that  y  alone  changes  in  /(ic,  y)j 

^u.^fjx,  y  +  Ay)  -/(a;,  y) 
Ay  Ay 

Now,  supposing  x  alone  to  change  in  (3), 

A  Mm\     /(x+Aa;,  y4-Ay)-/(a;4-Aa;,  y)-f(x,  y4-Ay)4-/(a;,  y).         /^x 
^x\AyJ  AX'  Ay  ^  ^ 


(1) 


(2) 


(3) 


Equating  (2)  and  (4),     ^^(£)=A(^;).      ' 

Hence  at  the  limits,         —  f  —  )=  —  f  —  V 
dy\dxj      dx\dyj 

In  the  same  manner  it  may  be  proved  that 
5®M  B^u  d^u 


dx^  dy      dy  dx^      dx  dy  dx 

This  principle  may  be  extended  to  any  number  of  differentiations, 
and  to  functions  of  three  or  more  variables. 

Art.  47.     Implicit  Functions. 

When  in  f{x,  y)  =  0,  y  can  be  expressed  as  an  explicit  function  of  a;, 
the  derivatives  may  be  found  by  the  methods  already  given.  In  tWs 
article  a  useful  formula  is  established  for  obtaining  the  first  derivative 
of  an  implicit  function. 

Let  i^=/(a.',  y)  =  0.  (1) 

Then  by  Art.  44,  Cor.  1, 

dx      dx      dy  dx 


70  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

But  u  =  0,  and  therefore  its  total  derivative  equals  0 ;  hence 

!^  +  |^^  =  0.  (3) 

ax      ay  ax  ^ 

solving  (3)  for  |,  gives     |  =  -|  (4) 

For  example  take  a^  +  2  2/a;  +  r^  =  0. 
Then  u  =  a?-{-2yx  +  r^. 

p  =  2x  +  2y,      p  =  2x. 
ax  ay 

Therefore  by  (4),    ^  =  -  ^^±^  =  -  ^±^. 
^  ^  ^'    dx  2x  X 

However,  when  an  implicit  relation  between  x  and  y  is  given  from 
which  y  cannot  readily  be  expressed  as  an  explicit  function  of  x,  it  is 
not  necessary  to  resort  to  the  method  just  given.  But /(a?,  y)  =  0  may 
be  immediately  differentiated  with  respect  to  x,  treating  t/  as  a  function 

of  X,  giving  what  is  called'  the  first  derived  equation,  from  which  —- 

dx 

can  be  obtained  as  a  function  of  x  and  y.     For  instance,  given : 

07^  —  3  axy  -|-  3/^  =  0. 
The  first  derived  equation  will  be 

3a^-3a2/-3aic^4-32/'^  =  0.  (1) 

dx  dx 

^Solving  (1)  for  ^,  ^^^-ay 

^  ^  ^        dx  dx     ax  -  f 

PROBLEMS. 

/     ,     V  .p      a^u        5V 

1.   «  =  oos(x  +  2,);  ■         venfy— =  ^-^. 

o     —  ^!_ihJ^.  'f     ^^^  —  ^^^ 

2/^  —  a^ '  dxdy      dy  dx 

8.   w  =  arc  tan  ( ?^ ) ;  verify  —^  =  — -^. 

\xj  dy^dx      dxdy^ 


/J   V  I 


IMPLICIT   FUNCTIONS.  71 


5.    w  =  sin  (aa;"  +  ft?/") ;  verify 


dx^  By  dz 


dx^  dy^      dif  dx^ 


6.   f-2m^y  +  af-a  =  0.  ^  =  ??^^^,  and  g  =  ^M^. 

ax     y  —  mx  dxr      {y  —  mxy 


3  2/  +  a;  =  0. 


dy  ^        1 

dx     3  (1  -  2/2)' 


Art.  48.     Integration  of  Functions  of  Two  or  More  Variables. 

Since  integration  is  the  inverse  of  differentiation,  a  partial  deriva- 
tive is  integrated  by  reversing  the  process  of  differentiation.  j     . 

For  example,  the  integral  of  ■—  =f(x,  y)  is  found  by  integrating 

ox^ 

twice  with  respect  to  x,  regarding  y  as  constant ;  but  as  y  is  regarded 
as  a  constant  in  this  integration,  it  must  be  noticed  that  the  constant 
of  integration  is  an  arbitrary  function  of  y. 

d^u  ■  ■'    '     ' 

Again,  let  it  be  required  to  integrate =/fe  y)- 

dydx^ 

This  may  be  expressed 

Evidently,  in  the  second  differentiation,  — -  was  differentiated  with 

dx 


reference  to  y  regarding  x  as  constant ;  therefore 

|=//(x,.)d,.  (2) 

In   (2),   ^l  is   evidently  such   a   function  that  its  derivative  with 
respect  to  a;  is   I  f{x,  y)  dy ; 


72  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

therefore,  ^  ~  I      I  /  (^?  2/)  ^2/  U^^> 

or  u  =jjf{x,  y)  dydx. 

In  Art.  46,  it  was  proved  that  the  values  of  the  partial  derivatives 
are  independent  of  the  order  in  which  the  variables  are  supposed  to 
change,  hence  the  order  of  integration  is  also  immaterial. 

5^ 


Similarly,  if 

.        eye:ey-^^^'y^' 

then 

u  "^ffffi^y  y)  dydxdy; 

and  if 

Jaler^^^'^'^^' 

then 

"  ^SSSS^^'^'  ^'  *)'*^'*2/* 

ff 

PROBLEMS. 

1.  Cfy^'^^^-- 

2 

■W-         •■  £££-' 

Art.  49.     Integration  of  Total  Differentials  of  the  First 

Order. 

If  tfc  be  a  function  of  x  and  y,  by  Art.  43, 

du  =  ^^dx-^^dy.  (1) 

dx  dy      .  ^ 

And  from  Art.  46,  #  ^^^ V  #  (^/)'  (2) 

Therefore,  if  a  total  differential  of  a  function  of  x  and  y  is  given  of 

the  form 

du  =  Pdx-^Qdy,  (3) 

then  P=^,   and    Q=^, 

ax  oy 

Hence  from  (2)  ^=^'  (4) 

dy       dx 


y^' 


CHANGE   OF   THE   INDEPENDENT  VARIABLE.  73 

which  is  the  condition  that  must  be  satisfied  to  make  (3)  an  exact 
differential.  This  condition  is  called  Euler's  Criterion  of  Integrability. 
When  (4)  is  satisfied,  (3)  is  an  exact  differential  of  a  function  of  x 
and  y.  Then  the  function  u  is  obtained  by  integrating  either  term  of 
(3);  thus 

i^=Jpdx +/(?/).  (5) 

In  (5),  the  integration  is  with  respect  to  x^  hence  the  constant  of 
integration  is  an  arbitrary  function  of  the  variable  which  is  treated  as 

a  constant,  and  /(y)  must  be  determined  so  as  to  make  —  =  Q. 

by 

For  example,  let        du  =  2  x^f^dx  +  3  x^yHy. 
Here  P=2xif,  and  Q^Sx^f. 

Hence,  ^=6xy\  and  ^^Gxf. 

oy  ox  ^ 

Therefore  (4)  is  satisfied,  and  (5)  gives 

u=  i  2 xy^dx  =  x^y^  +f{y)  =  ^1^  +  c. 

PROBLEMS. 

1.  du  =  ydx-\-xdy.  w  =  icy  +  c. 

2 .  du  —  ^  a^y^dx  +  3  x*y^dy.  u  =  a; V  +  c. 

3.  du  =  j-{-(2y-^^dy.  u  =  ^-^y'-\-c. 

Art.  50.     Change  of  the  Independent  Variable. 

Hitherto,  the  derivatives  — ,  — ,  etc.,  have  been  obtained  on  the 

dx  dx- 

supposition  that  x  was  the  independent  variable  and  y  the  function, 
but  it  is  sometimes  advantageous  to  change  the  function  into  another 
one  in  which  y  is  made  the  independent  variable  and  x  the  function. 
And  occasionally  it  is  desirable  to  make  a  new  variable,  of  which  both 
x  and  y  are  functions,  the  independent  variable. 


74 


DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 


(a)  To  express  —  in  terms  of  — • 
dx  dy 

If  2/  is  a  function  of  x,  then  x  may  be  regarded  as  a  function  of  y, 
and*y  may  be  treated  as  the  independent  variable.     Evidently 

Ay      Ax      ^ 
Ax     Ay 

and  as  Ay  approaches  zero,  Ax  approaches  zero,  and  at  the  limit, 

^x  — =1- 
dx     dy 

dy_l_  •     '  '    • 

therefore  dx  ~  dx  (1) 

dy     ■    " 


(6)  To  express  — ^  in  terms  of  —  and  — -,  also  to  express  —  in 

^    '  rl^  fill  rl-ifi  •"■  /7/m3 


dx" 

. „    o  dx  d^x -I  d^x 

terms  of  — ,  --,  and  — -• 
dy   dy^  dy^ 


From  (1), 

By  Art.  44,  Cor.  3, 


dy  dy^ 


d^ 


d^^±fdy\ 
d3?     dx  \dxj 

d^_±  (  1 
dot?  ~~  dx  I  dx 


therefore 


d 

1 

d 

1 

dx 

dx 

dy  . 

~dy 

dx 
dy) 

d'y      d 
dx'~'dy 

1 

'dx 

dy} 

d^ 

dy^       dy 
dx 


dx 


dx 


dPx 
dy^ 


fdxV     dx         fdxV 


(^) 


CHANGE  OF  THE  INDEPENDENT  VARIABLE. 


75 


Similarly,       ^  = 


d 
dx 

~  d-x  1 
df 

/dx\^ 

d 
dy 

r  d'x  ' 

dy' 

fdxV 

\dyj_ 

dy 
dx 

V 

ixVd'x     ^fdxVfd'xY 
lyj  df        [dyj  W) 

dy 

m 

dx 

[dyjdf        W) 

_i 

fdx' 

/ 

(3) 


In  equations  (1),  (2)  and  (3),  the  independent  variable  is  changed 
from  xtoy. 

(c)   To  express  -^,  — ^,  etc.,  in  terms  of  -^,  J,  etc.,  when  x  is 
dx    dx^  dz    dz^ 

sdme  given  function  of  2;.  .  .  , 

By  Art.  44,  Cor.  3, 

dy  _dy  dz^ 
dx      dz  dx^ 

therefore  ^  =  --  ("^^  =  —  (^  —  • 

dx^     dx\dxj      dz\dxjdx^ 

and  ^^±f^dz_ 

dx^      dz\dx^Jdx 

In  equations  (4),  (5)  and  (6),  the  independent  variable  is  changed 
from  a;  to  2;. 


(4) 

(5) 
(6) 


PROBLEMS. 

1.   Change  the  independent  variable  from  a;  to  ^  in 

dx^      \dxj      dx 

Substituting  the  values  of  ^„  and  ^  from  (1)  and  (2), 

dar  dx 


76 


DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


X 


therefore 


df_ 

dy 
d'x 


+ 


1  + 


ir-i=o: 

dx\       dx 


dy 


fdx\^ 


dy 
=  0. 


d/  \dy^ 

2.   Change  the  independent  variable  from  a:  to  2  in 


when  X  =  cos  z. 

By  (4), 


and 


^  ^dx^       dx       ' 


dy  _dy  dz  ^ 
dx     dz  dx^ 

^  =  -sin2,  hence  ^  =  — A-> 
dz  dx         sins 

dy_ 1_  d^ 

dx         sin  z  dz 


By  (5), 


d^y  _  d^  fdy\  dz  ^ 
dx^     dz  [dxj  dx ' 


Therefore 


dz\dxj     dz\     sin  zdz  J 

_  cosg  dy  _    1     d^y 
sin-  z  dz      sin  z  dz^ 

d^y  _  _  /cos  z  dy  _    1     d^y\    1 
dar^         vsin^zc?2;  -  sinz  c?2^/sin« 


—  _  /"CQSZ  d^  _     1     dy 
\sin^  z  dz      sin^  ^  dz^ 


) 


(7) 


(8) 


Substituting  the  values  of  -^  and  -4  from  (7)  and  (8)  m  the  given 

dx  dar 

example,  gives 


-(1 

Hence 


\sin^  z  dz      sin^  z  dz^J  \     sin  z  dz) 


^y 


dz^ 


ra. 


CHANGE  OF   THE  INDEPENDENT   VARIABLE.  77 


3.    Given  y  =/(«),  and  x  =  F(t),  to  express  -^,  and  — ^  in  terms  of 
^,  ^,  %  and  ^. 


5 
(«r  dt^'  at'  "~"  df 


B,A.t.44,Co.3,  1  =  11  (1) 

Differentiating  (1),  and  treating  -^  as  a  function  of  t  through  a?,  gives 

dx 

^_d^do^     dy  d^ 
df~  da^df      dxdt^' 

d^^dy^^     d^y  dx     fe  dy 
„  d*w     d^     dic  df      df  dt      df  dt  ,ox 

^^^^^  S?  = d^ = dl (^) 

dt;'  df 


-      '  CHAPTER   VIII.    ,  / 

DEVELOPMENT  OF  FUNCTIONS. 
Art.  51.     Definition. 

A  Function  is  said  to  be  developed  when  it  is  transformed  into  an 
equivalent  series. 

By  the  Binomial  Theorem,  constant  powers  of  a  binomial  can  be 
developed  into  series.     For  example, 

Some  fractional  functions  may  be  developed  by  actual  division. 
For  example, 


l-3aj 


The  Calculus  method  of  development  is  a  general  method,  including 
the  developments  just  given  and  many  others  as  special  cases. 

This  is  one  of  the  most  important  applications  of  successive  derivar 
tives. 

Art.  52.     Maclaurin's  Theorem. 

Maclaurin's  Theorem  is  a  theorem  by  which  a  function  of  a  single 
variable  may  be  developed  into  a  series  of  terms  arranged  according 
to  the  ascending  integral  powers  of  that  variable,  with  constant  co- 
efficients. 

The  function  to  be  developed  is 

Assume  the  development  of  the  form 

y  =f(x)  =  A-^Bx-{-Cx'-^Da^  +  E^-*-  (1) 

78 


DEVELOPMENT  OF  FUNCTIONS.  79 

in  which  A,  B,  C,  D,  etc.,  are  constants  to  be  found  by  the  me,thod  of 
Undetermined  Coefficients, 

Forming  the  successive  derivatives  of  (1) : 

^=B-\-2Cx+      SDa^+           4jE;a^+...  (2) 

dx  ^ 

^=         2C+2'3Dx-h      3.4^x-2+-  (3) 
ux^ 

^=                     2'3D    -i-2'S'4.Ex  -{-"'  (4) 


Since  (1)  and  consequently  (2),  (3),  etc.,  are  assumed  to  be  true  for 
all  values  of  x,  they  will  be  true  when  »  =  0.  Hence,  making  x  =  0 
in  each  of  these  equations,  and  representing  what  i/  becomes  on  this 

hypothesis  by  (y) ;  what  -^  becomes  by  f-^] ;  what  J  becomes  by 
/^2  \  dx  \dxj  dx^ 

'  — ^  J ;  and  so  on :  there  follows 

from  (1),       (2/)      =  A,  or  A  =  (y)  ; 


Substituting  these  values  of  A,  B,  C,  •••  in  (1),  gives 

,=/(.)  =  (,)  +  (g.,.(3)|V(g)|+...  (5) 

If  the  function  and  its  successive  derivatives  are  expressed  by 

f{x),  n^),  r{x),  f'%x\  etc., 

equation  (5)  may  be  written, 


80  DIFFERENTIAL  AND   INTEGRAL  CALCULUS. 

•  •      y  =/(^)  =/(0)+/'(0)  I +/"(0)  I +/"'(0)|  +  -,        (6) 

^vilich  is  the  formula  of  Maclaurin's  Theorem.* 

A  If   in   the  attempted  development  of   a  function   by  Maclaurin's 

j       Theorem,  the  function  or  some  one  of  its  derivatives  becomes  infinite 

when  x  =  0,  the  function  cannot  be  developed  by  Maclaurin's  Theorem. 

j     This  is  evident,  because  a  finite  function  cannot  be  equal  to  a  series 

/     containing  infinite  terms. 

^  PROBLEMS. 

1.   To  develop  2/=  (<^  + a;)". 
Here 

f{x)    =(a  +  a;)";  hence /(O)    =a". 

f{x)  =w  (a  4- «)"-';  "      /'(O)  =wa"-^ 

fix)  =  n{n  -  l)(a  +  xy-^-,  «      /"(O)  =  n{n  -  l)a^-\ 

f"'(x)=:  n(n-  l)(n  -  2)(a  +  xy-' ;  "      /'"(0)=  n(n  -  l)(7i  -  2)a' 


n-3 


Substituting  in  (6),  Art.  52, 

y=(a  +  xy=a^-^na^-'x-^''^''-^)a--'x^  +  ''(''-}^(''-^K»-^a^-^..., 

\2  [3 

which  is  the  same  development  as  that  given  by  the  Binomial  Theorem. 
2.    To  develop  2/ =  log  (1  +  a;).  . 

Here,  f(x)    =  log  (1  +  ») ;   hence  /(O)    =  0. 


I  ^  M"  -^  ^"^  =rT^' 


m 


r(^)=-7T-^^T-.;     "   /"W  = 


(l+o:) 
■2.m 

(1  +35)^' 


f"(x)=  \'  ^  •  ^:  "      /"(0)=  1.2m. 


*  This  theorem  is  commonly  known  as  Maclaurin's,  having  been  first  published 
by  him  in  1742  ;  but  as  it  had  been  given  in  1717  by  Stirling,  it  should  more  prop- 
erly bear  the  name  of  the  latter. 


DEVELOPMENT   OF    FUNCTIONS.  81 

Substituting  in  (6),  Art.  52,  gives 

y  =  \og(l  +  x)=m(x-'^  +  ^-^  +  ...\ 
and  if  the  logarithm  is  in  the  Naperian  System, 

log(l+.)=x-f  +  f-}+-..     . 

Thus  the  logarithmic  series  is  found  to  be  a  special  case  under 
Maclaurin's  Theorem. 

3.    To  develop  y  =  sin  x. 

f(x)    =  sin  a; ;  hence  /(O)    =  0. 

f'(x)   =cosa;;  "      /'(O)   =1. 

f"(x)=-smx',  "      /"(0)=0. 

f"'(x)  =  -cosx',  "      /'"(0)=-~l. 

Therefore,  3,  =  sina;  =  05-^  +  ^-^  + .... 

i£     [^     LL 

In  the  successive  derivatives  of  sin  a;,  the  first  four  values  are  periodically- 
repeating ;  i.e.,  the  fifth  derivative  equals  the  first,  the  sixth  equals  the  second,  etc.; 

hence,  in  general,  d'^Csm  x)  _  gjn  f  ^  +  w  -  V 

Obtain  by  Maclaurin's  Theorem  the  following  developments : 
^      ^.4      ^6 

4.      COSiC  =  l—  ; . 

^  \±  ^ 


The  general  formula  for  the  successive  derivatives  of  cos  x  is 

d"(cosa;) 


cos  (  x  +  w  ^-  J  ■ 


By  the  aid  of  the  last  two  developments,  natural  sines  and  cosines 
may  be  computed. 

For  example,  to  find  the  sine  of  45°. 

By  Art.  22,  the  circular  measure  of  45°  is  -•  Substituting  this 
value  of  x  in  the  series  of  Prob.  3,  gives 

=  .7071068. 


82  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

5.  a-  =  l  +  loga^  +  log2a^4-log«a^4--. 

1  ^  [^ 

If  a  =  e  and  a;  =  l   in  this  series,  the  value  of  the  Naperian  base 
may  pe  computed. 

/y»3  /ytO  /yj 

6.  arctana;  =  aj  — — +  — —  77  H . 

o      5      7 

The  labor  in  finding  the  successive  derivatives  may  sometimes  be 

lessened  by  expanding  the  first  derivative  by  some  one  of  the  algebraic 

methods,  as  follows : 

* 
f(x)  =tan~^a;. 


By  substituting  cc  =  1  in  the  development  of  arc  tan  Xj 

4  3^6      7^ 

By  Trigonometry,  arc  tan  1  =  arc  tan  ^  +  arc  tan  ^. 

"-i=[i-l©'-KIJ-]-[i-i(IJ*KiJ--} 

Therefore  ir  =  3.1415924-. 

/v«2  /y»4  n^ 

8.  e"°*  =  l4-a;+ --^ ^-.... 

2      2-4     3-5 

9.  e*seca;=l4-a;  +  a^4-^H • 

o 

11.   arcsm^  =  .,  +  273  +  2:^  +  2:4:^  +  -. 

It  was  by  the  use  of  this  series  that  Sir  Isaac  Newton  computed  the  value  of  v. 


DEVELOPMENT  OF   FUNCTIONS.  83 

12.  Develop  y  =  logic. 

f(x)    =  log  X ;  hence  /(O)    =  —  oo. 

f'(x)  =        -;  hence /'(O)  =      oo. 
f"(x)  =  -  ^ ;  hence  f"(0)  =  -  oo. 

Substituting  in  Maclaurin^s  Formula,  gives, 

y  =  logx  =  —  cc  +  QO^-QO  — H . 

In  this  example,  \ogx  equals  a  series  of  terms  involving  oo,  which 
makes  the  development  indeterminate  for  all  values  of  x. 

Hence  this  function  cannot  be  developed  by  Maclaurin's  Theorem. 

13.  Develop  y  =  cot  x. 

14.  If  e  be  substituted  for  a  in  Ex.  5, 

Substituting  a;  V— 1  for  a;,  . 

e^^~i==l  — — -f -— —  H uV^/^a;  — — +  — —  —  4----^ 

[2+^     [6+       +^     ^'^     |3+[5     [7+     J 

=  COS  a;  +  V—  1  sin  a;,  by  Exs.  4  and  3.  (2) 

Substituting  —  x  V—  1  for  x  in  (1),  gives  similarly 

g-^^-^  =  cosa;— V— 1  sinar.  (3) 

Combining  (2)  and  (3),  gives 

sin  a;  = ^ — , 


2V-1 

and  cosa;  = '— 

These  values  of  the  sine  and  cosine  are  called  their  exponential 
values. 

The  real  functions,  ^"^  ~  ^  "^  and  ?f_+_l^,  are  called  respectively  the  hyperbolic 
sine  and  hyperbolic  cosine  of  x  and  are  written  sinh  x  and  coshic. 


84  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Art.  53.     Taylor's  Theorem. 

*  Taylor's  Theorem  is  a  theorem  for  developing  a  function  of  the 
sum  of  two  variables  into  a  series  of  terms  arranged  according  to  the 
ascending  powers  of  one  of  the  variables,  with  coefficients  that  are 
functions  of  the  other  variable. 

Taylor's  Theorem  depends  on  the  following  principle  which  must 
first  be  established :  The  derivative  of  a  function  of  the  sum  of  x  and 
y  with  reference  to  x  regarding  y  as  constant,  is  equal  to  the  derivative 
of  the  function  with  reference  to  y  regarding  x  as  constant. 


Let 

w=/(a;  +  2/). 

Substituting 

z  —  x-{-y,  gives 

u  =/(.). 

In  the  first  case, 

du_df(z)  dz 
dx        dz     dx 

=/'(.),  since  1^=1. 

In  the  second  case, 

du  _  df(z)  dz 
dy        dz     dy 

=/'(2),  since  ^  =  1. 
dy 

Therefore 

du      du 
dx  =  dy' 

This  principle  may 

readily  be  shown  to  apply  in  any  particular 

example ;  for  instance, 

let 

u  =  (x-\-yy, 

then 

dx      dy                 ^ 

Art.  64.     Demonstration  of  Taylor's  Theorem. 
Let  u=f{x-\-y). 

Assume  the  development  to  have  the  form 

u=f{x^y)=A  +  By-^Cf-\-Df-^'*',  (1) 


DEVELOPMENT  OF  FUNCTIONS.  85 

in  which  A,  B,  (7,  •  •  •  are  independent  of  y,  but  are  functions  of  x.  It 
is  now  required  to  find  values  of  A,  B,  C,  •••  by  the  method  of  un- 
determined coefficients. 

Differentiating  (1),  first  with  reference  to  x,  regarding  y  as  constant, 
then  with  reference  to  y,  regarding  x  as  constant, 

dx      dx      clx^       dx  dx  * 

^=B  +  2Cy-{-3Df-^4:Ef  +  .^.. 
dy 

But  by  Art.  53,  —  =  — ;  therefore, 
dx      dy 

^+^.y+^^^  +  ^/  =  5  + 2  0?/ +  32)2/2  +  4^/+....     (2) 
dx       dx         dx  dx  ^  v  / 

Making  y  =  0  in  (1),  gives  A  =f(x). 

Since  (2)  is  true  for  every  value  of  y;  equating  the  coefl&cients  of 
like  powers  of  y  in  the  two  members  by  the  principle  of  Undeter- 
mined Coefficients, 

^  =  B,       henceB  =  ^         .      =/'(.), 
f  =  2C,     hence  C  =  l|(/(.)         =l/"(x), 

f=3A    hence  Z>  =  ||(i/"(.))  =  i/"'(.);  etc 

Substituting  these  values  of  A,  B,  C,  •••  in  (1),  gives 

u=f{x-{-  y)  =f(x)  +r(x)  y  +f\x)  ^'  +/"'(x)  t  +  ...,  (3) 

which  is  Taylor's  Theorem. 

If  x=  0  be  substituted  in  (3),  it  reduces  to 

m  =/(0)  +/'(0)  y  +  /-"(O)  t  +/'"(0)  2^  + ..., 

If  12 


86  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

which  is  Maclaurin's  Theorem.  So  Maclaurin's  Theorem  may  be  con- 
sidered as  being  but  a  special  case  of  the  more  general  one,  Taylor's 
Tneorem.* 

PROBLEMS. 

1.  To  develop  (x  -j-  ?/)". 

Substituting  y  =  0,  and  taking  the  successive  derivatives, 
f(x)     =x% 
f'(x)    =na;"-i, 
f"(x)  =n(n-l)af'-^, 
f"'(x)  =  n(7i-  1)  {n  -  2)  a^-3. 

Substituting  these  values  in  (3), 
(x  +  yy  =  x«  +  nx^-'y  +  ^(^~^)  x^'Y  +  ^(^-l)(^-2)  ^n-SyS  +  ...^ 

which  is  the  Binomial  Formula. 

2.  To  develop  sin  (a; -f- 2/). 

/(aj)    ==  sin  X,        f'(x)    =  cos  x, 
f"(x)  =  —  sin  X,    /'"(aj)  =  —  cos  x. 

Therefore  sin  (a;  +  2/)  =  sin  aj/^l  —  -|  +  -|- j  +  cos  xfy  —  ^  -f  ^ j 

=  sin  X  cos  y  +  cos  x  sin  y. 
Obtain  the  following  developments  by  Taylor's  Theorem : 

S.   a=^+y  =  a'(l  +  loga  •  ?/  +  log^  a^+  log^a^^  -.) 

If.  l£ 

4.  Iog(a;  +  2/)  =  logaj-f ^-^^-f--^-.... 

5V    -ry;         8     t  ^     2  ar^     3  a:^ 

5.  (a;  +  y)^  =  a;^  -f  Ja;"^y  -  ^aj'V  +  A^"V • 

♦Taylor's  Theorem  is  named  from  its  discoverer,  Dr.  Brook  Taylor.  It  was 
first  published  in  1715,  in  a  book  by  Dr.  Taylor  entitled  Methodus  Incrementorum 
Directa  et  Inversa. 


DEVELOPMENT  OF  FUNCTIONS.  87 


6.  log(l  +  sm*)  =  x-|  +  |-^+-. 

7.  log  sec  (x  +  y)  =  log  sec  x  +  tan  x-y  -{■  sec^  x  •  ^ 

4-sec^a;  •  tana;*  ^.... 
3 


Art.  55.     Rigorous  Proof  of  Taylor's  Theorem. 

In  the  demonstrations  of  Taylor's  and  Maclaurin's  Theorems,  it  was 
assumed  that  the  development  would  take  place  in  a  proposed  form, 
and  an  infinite  series  was  used  without  ascertaining  that  it  was  con- 
vergent. On  account  of  these,  as  well  as  other  objections,  the  method 
used  is  not  altogether  satisfactory.  But,  on  the  other  hand,  a  rigorous 
investigation  is  necessarily  complex  and  indirect.  The  proof  which 
follows  is  one  of  the  least  difficult  ones. 

The  following  proposition  must  be  first  established : 
If  (^  (a;)  =  0,  when  x=a,  and  also  when  x  =  h,  and  if  <\>  {x)  and 
<^'(x)  are  finite  and  continuous  between  these  values;  then  <l>\x)  will 
vanish  for  some  value  of  x  between  a  and  h. 

The  limit  of  — ^  =  -^,  and  hence  -^  will  have  the  same  sign  as  -^ 
Ao;     dx  dx  ,      ^x 

Ay 
when  Aa;  is  taken  small  enough.    If  y  increases  as  x  increases,  t-  will 

be  positive,  and  if  y  decreases  as  x  increases,  —  will  be  negative.     So, 

dy 
if  "T-  is  always  positive  between  the  two  given  values  of  x,  ^  {x)  would 

be  constantly  increasing,  and  if  -^  is  always  negative  between  the  two 

values  of  x,  (ji  (x)  would  be  constantly  decreasing ;  but  neither  suppo- 
sition can  be  true,  as  <^  (x)  vanishes  at  the  two  given  values  for  x. 

Therefore,  <j>'(x)  must  change  its  sign  between  the  two  values,  but 
a  variable  can  only  change  its  sign  by  passing  through  zero  or  infinity, 
and  4>'(x)  remains  finite  by  hypothesis;  hence,  <^'(a;)  must  pass  through 
the  value  zero. 

Let  f{x)  and  its  successive  derivatives  be  finite  and  continuous 
between  x  =  a,  and  x  =  a  -\- h. 


88  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

Assume 
*  (X)  =f{a  +  X)  -/(a)  -  xf'io)  -  |/"(a) ...  -~f{a)  -  r^S,    (1) 


in  -vj^hich 

■rf(a  +  k)-f(a)-hf{a)-^f"{a)...  -|/"(«)1-       (2) 


o_l!L+l| 


In  (2),  it  is  to  be  observed  that  E  is  independent  of  x. 
In  (1),  it  is  evident  that  cf>{x)  =  0  when  x  =  0,  and  when  x  =  h. 
Hence,  <^'(^)  iii^st  be  equal  to  zero  for  some  small  value  of  x  between 
0  and  h.     Represent  this  value  by  x^. 

Taking  the  derivative  of  (1)  with  respect  to  x, 

.t,'{x)=r(a  +  x)-f'{a)-xf"(a)-  p'"{a) ...  -  -^/-(a)  -fs    (3) 

Ir  |yi  —  1  [n 

=  0,  when  x  =  x^. 

But  (3)  also  vanishes  when  a;  =  0 ;  hence  there  is  some  value  of  x 
between  0  and  ajj,  for  which  <^"(a;)  =  0. 

Continuing  this  process  to  w  +  1  differentiations, 

<l>-+\x)=p^\a  +  x)-Ii, 

for  some  value  of  x  between  0  and  h,  <f>''+\x)  =  0 ;  let  this  value  of  x 
be  Oh,  where  0  <1,  therefore 

r^\a  +  ek)  =  R.  (4) 

Equating  the  values  of  E  in  (2)  and  (4),  and  solving  for  /(a  +  h), 

f(a-\-h)  =  f(a)+hf'(a)  +  ^f"(a)-.  +,-/"(«)  + ,-^/"^^(a  +  ^/O-    (5) 

Now,  since  the  only  restriction  imposed  on  a  was  that  it  must  be 
finite,  a  may  have  any  value ;  hence  x  may  be  substituted  for  a  in  (5), 
which  gives 

f(x  +  h)=f(x)+hfXx)+^^f"(x) ...  -^ffn(x)  +  -^r-\x+eh).    (6) 
From  ^0),  Taylor's  Theorem  follows  whenever  the  function  is  such 


DEVELOPMENT   OF   FUNCTIONS.  89 

that  by  sufficiently  increasing  n  the  last  term  can  be  made  indefinitely 

small.* 

Art.  56.     Remainder  in  Taylor's  and  Maclaurin's  Theorems. 

The  last  term  of  (6),  Art.  55,  -^ — -f'-^x  +  Oh),  is  called  the  re- 
mainder after  n  +  1  terms.  

For  example,  \etf(x)  =  (1  +  ic)™,  then  by  (6),  Art.  55, 

(1  +  ic)™  =  1  -\-  mx-\ ^— ^x^  H-  •  •  • 

+    ^"^'   [m(m-  l)...(m  -n)(l  -f- ^a^r"""^'!. 
|n  +  1  ^         -' 

/pn  +  l 

In  this  development, [??i(m  —  1)  •••  (m  —  n)(l  +  6'a;) '""""*]   is 

the  remainder.  ' 

If  X  is  less  than  1,  the  last  term  can  be  made  indefinitely  small  by 
sufficiently  increasing  m. 

Hence,  when  x<l,  1  +  mx  +  ^  ^^i o~ — ^^  + '"  is  a  convergent 
series.  - 

If  x  =  0  be  substituted  in  the  remainder  in  Taylor's  Theorem,  and 
then  X  be  substituted  for  h,  the  remainder  in  Maclaurin's  Theorem  is 
obtained,  which  is 


|n-f  1 


r^'iox). 


If  this  remainder,  when  n  is  taken  sufficiently  large,  becomes  indefi- 
nitely small,  Maclaurin's  Theorem  gives  a  convergent  series. 

For  example,  let  f(x)  =  sin  x. 

Then   sin(x)  =  a;-|  +  |-|  +  ...-g^sm|^ea:  +  (n  +  l)|j 

But  whatever  may  be  the  value  of  x,  it  is  evident  that 

,n+l 


X' 

sm 


In  +  l 


Ox +  (71-^1)^ 


*  The  proof  of  Taylor's  Theorem  given  in  this  article  is  due  to  Mr.  Homersham 
Cox. 


90  DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 

will  have  zero  for  its  limit,  hence  this  series  is  convergent  for  all  values 
of  a;. 

Art.  57.     Taylor's  Theorem  for  Functions  of  Two  or  More 
Independent   Variables. 

Let/(a;,  y)  be  a  function  of  two  independent  variables,  and  suppose 
f(x  +  h,  y  -\-k)  \^  to  be  expanded  in  ascending  powers  of  h  and  h. 

Regarding  y  as  constant,  and  expanding  as  though  x  was  the  only- 
variable, 

/(^4-^,2/  +  ^)=/(^,  2/  +  ^)+^£/(^.2/+^')  +  |^/Gx-,2/  +  ^-)  +  -.     (1) 

Expanding  f{x,  y  -f  k),  regarding  x  as  constant,  and  y  as  the  only- 
variable, 

f{x,y  +  h)=f{x,  y)  +  kl-f{x,  J,)  +1  J^/(x,  ,j)  +  .:.  (2) 

Substituting  this  value  of /(.r,  >/  +  k)  from  (2)  in  (1),  gives 

fix  -{'h,y-f-  k)  =  f(x,  y)  +  h  ~f{x,  y)  +  k  -/{x,  y) 

ox  oy 

Similarly, 

f(x  -\-h,y  +  k,z  +  l) 

=  f(x,  y,  z)  -f  h  -^/(x,  y,z)-\-k  |- /(aj,  y,z)+l  |/fe  y,  ^) 


4- ir^'^/C^'^  2/,  2)  +  3 «  ^/(a^,  2/,  ^)  +  ..•  1+ 


And  in  like  manner  a  function  of  any  number  of  independent  vari- 
ables may  be  expanded. 


CHAPTER   IX. 

EVALUATION  OF  INDETERMINATE  FORMS. 

Art.  58.     Indeterminate  Forms. 

A  function  of  x  is  indeterminate  when  the  substitution  of  a  par- 
ticular value  for  x  gives  rise  to  one  of  the  following  expressions : 

^,    ^,00-00,    0°,    00°,    1^*. 


0 


CO 


The  true  value  of  a  function  which  becomes  indeterminate  is  the 
value  which  the  function  approaches  as  its  limit,  as  the  independent 
variable  approaches  the  particular  value  which  makes  the  function 
indeterminate. 

For  example,  to  find  the  true  value  of  — ^^  when  x  =  a. 

X  —  a 

When  £c  =  a,  this  fraction  assumes  the  form  -• 
If  a  +  h  is  substituted  for  x,  the  fraction  becomes 


(ci-^hr 


a  -{-  h  —  a 


=  2a-^h. 


Now  if  h  approaches  zero,  the  independent  variable  approaches  the 
particular  value  a,  and  the  function  evidently  approaches  2  a  as  its 
true  vahie. 

Again,  if  both  numerator  and  denominator  of  the  fraction  ^  ~^ 

X  —  a 
are  divided  by  x  —  a,  the  quotient  is  x -{- a,  and  now  when  x  =  a,  the 

true  value  is  found  as  before  to  be  2  a. 

As  another  example, 


ar  0 

91 


92  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

By  rationalizing  the  numerator, 


a  —  Va^  —  ^  _ 

X 


XT 


a-  -  (g^  -  x")  ^  r       1       n    ^  \_ 


By  algebraic  and  trigonometric  transformations  the  true  values  of 
some  indeterminate  forms  can  be  readily  found,  but  the  Differential 
Calculus  furnishes  a  method  of  very  general  application. 


Art.  59.     Functions  that  take  the  Form  -• 

Let /(a?)  and  <^{x)  be  two  functions,  such  that  f(po)=  0  and  <f)(x)  =  0, 
when  a;  =  a ; 

then  m  =  ^-. 

<l>(a)     0 

Let  X  take  an  increment  h ;  then  by  Taylor's  Theorem, 


..    „    f(^)+f'i^)j^+f"(^)^+f"'i^)^+- 

f(x  +  h)  ^ [2 \3 


(1) 


Substituting  a  for  x,  making  f(a)  =  0  and  <^  (a)  =  0,  and  dividing 
both  terms  of  the  fraction  by  h, 

f{a  +  K)  ^ l£ i£  m. 

Hence,  as  ^  approaches  zero,  by  Art.  5, 

<t>(a)     <l>'{a)'  ■  ^' 

fix) 
which  is  the  true  value  of  Vr^  when  x  —  a. 

<f>(x) 

If  /'(a)=0,  and  <^'(a)=0,  then  '-^^  =  ^,  and  the  result  is  still 
indeterminate.     In  this  case,  dropping  the  first  term  of  the  numerator 


EVALUATION  OF  INDETERMINATE  FORMS.  93 

and  also  of  the  denominator  of  (2),  dividing  both  terms  of  the  fraction 
by  — ,  as  h  approaches  zero, 

</,(a) -.#,"(«)'  ^^ 

If  (4)  is  also  indeterminate,  the  process  is  repeated  until  a  ratio  of 
two  derivatives  is  obtained,  both  of  which  do  not  reduce  to  zero  when 
x=a. 

If  /"(a)  =  0  and  <l>\d)  be  not  0,  the  true  value  is  0. 

If  /"(a)  be  not  0  and  <f>"(a)  =  0,  the  true  value  is  x. 

PROBLEMS. 

ar*  —  1 

1.  Find  the  true  value  of -,  when  x  =  l. 

x  —  1 

f{x)=a^-l,  <f>{x)=x-l', 
hence,  f(x)=ba^j  and  <l>'{x)  —  1 ; 

therefore  4^  =  -Q^  =  ^  =  ^y  when  a;  =  1. 

<^{x)      <f>'(x)        1 

2.  Find  the  true  value  of ,  when  x  =  0. 

or 

4,(x)  ^  0'  ' 

fM   =lzi^^  =  0   whence  =  0; 

£M=?iM        =2,  whenc.  =  0; 
4,"{x)        6  a!  0'  ' 

rM^^J^        =1,  wlien=.  =  0; 
<i,"\x)        6  6'  ' 


tWore         [^^L  =  i. 


The  subscript  denotes  the  value  which  is  to  be  substituted  for  x  in  the  function 
within  the  brackets. 


94  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

3.  ^^,  when  x  =  l.  Ans.  1. 
X  —  1 

4.  ^^-^ — ,  when  x  =  0.  Ans.  log-* 
^    X  b 

5.    ,  when  a:  =  0.  Ans.  2. 

sinx 

6. ,  when  x  =  1.  Ans.  n. 

1—x 

_     tana;  +  secic  —  1  ,  ^  ah 

7. -,  when  x  =  0.  Ans.  1. 

tana;  —  seca;  +  r 

8.  — — ^ — ,  when  x  =  0.  Ans.  2. 

X  —  sm  X 

9. — ■ — ,  when  x  =  a.  Ans.  0. 

sr  —  a'^ 

10.  — — ^  ?^~^ — -„ :,  when  x  —  a.  Ans.  —  oo. 

-  -     tan  X  —  sin  x  i,  a  a        ^ 

11.    r—^ ,  when  x  =  0.  Ans.  4-. 

sin"^a; 


Art.  60.     Functions  that  take  the  Form  —. 

00 

When  a;  =  a,  a  function  may  take  any  one  of  the  forms 

^,    ±,    or    2^. 
a     cc  GO 

Evidently,  —=  cc,  —  =  0,  and  —  is  indeterminate, 
a  00  00 

Let  /^  =  ^,  when  a;  =  a. 

<^  {X)         00 

By  taking  the  reciprocals  of  f(x)  and  <^  (a;), 

1 


m_i(^  =  0   ^hena;  =  a. 
<f>(x)     J_     0' 


EVALUATION  OF  INDETERMINATE  FORMS.  95 

Therefore,  by  Art.  59, 

1         d  f'l   \  4,'(x) 

f{x)  _i>{x)^ dxKj. (x)J  [<!> ix)J     ^'{x) lf(x)J 

fix)       dx{j\x)J  lf(x)J 

and  when  «  =  a,  /W^^MIZW;  (1) 

hence  tMl^fM.  (2) 

4,(0)      <#.'(«)  ' 

Therefore,  the  true  value  of  the  indeterminate  form  —  can  be  found 
by  the  same  method  as  that  of  the  form  -. 

However,  in  dividing  (1)  by  :il— i,  it  vyas  assumed  that  Zi^  is  not  equal  to 
0(a)  0(a) 

0  or  00 .    But  (2)  gives  the  true  value  in  these  cases  also,  as  may  be  shown  as 

follows  :  ^  . 

Suppose  the  true  value  of  ~Ar  to  be  0,  and  let  ;i  be  a  finite  quantity  ;  then 
0(a) 

(PCa)  <p{a) 

i±M^has 
0(a) 
to  it,  giving 


But  /C^)+  h,(p{a)  jj^g  ^  value  which  is  neither  0  nor  oo ,  hence  (2)  will  apply 
0(a) 


f{d)+h<t>{a)  ^  f'(a)  +  h<p'(a) 
<t>{a)  V(a)         ' 

therefore  /(a)^/(a).        .. 

0(«)       0'(«) 

Similarly,  if  the  true  value  of  Z^^  =  qo  v^hen  jc  =  a,  then  JV^  =  0,  and  the 

0Ca)  f(a) 

same  demonstration  applies.  ^  -^  ^  ^ 

Art.  61.     junctions  that  take  the  Forms  0  x  oo  and  oo  —  x. 
Let  f(x)  X  <f>(x)  =  0  X  <X),  when  x  =  a. 

Then  f^a)x<f>(a)=^  =  l', 

therefore,  the  true  value  may  be  found  as  in  Art.  59. 


96  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

Again,  let         f{x)  —  (t>(x)  =  cc  —  co,  when  x  =  a. 

The  expression  in  this  case  can  be  transformed  into  a  fraction, 

*  0 

which  will  assume  either  the  form  -  or  — ,  and  the  true  value  is  found 

0       *^ 
as  before. 

For  example,  to  find  the  value  of 

sec  X  —  tan  x,  when  cc  =  -• 

2 

sec  X  —  tan  x  = =  -,  when  a;  =  -• 

cos  X         0  2 


Therefore  Ml   ^[^.^^1   _o. 


Art.  62.     Functions  that  take  the  Forms  0®,  oo^,  and  1±*. 

Let  f(x)  and  <^  (x)  be  two  functions  of  x,  which,  when  x  =  a,  take 
such  values  that  [/(»)]*''  is  one  of  the  assumed  forms. 

Let  y  =  lf(xW^', 

then  log 2/  =  <^  (x)  logf{x).  (1) 

1st.   When/(aj)  =  oo  or  0,  and  <f>(x)  =  0,  (1)  becomes 

<l>(x)\ogf{x)  =  0{±y,). 

2d.   When /(a?)  =  1,  and  <^(x)  =  ±  oo ,  (1)  becomes 

c}>(x)logf(x)  =  (±oo)xO. 

Therefore,  the  true  values  of  the  logarithms  of  all  the  functions 
which  take  the  forms,  0*^,  oo°,  and  1-°^,  may  be  obtained  as  in  Art.  61. 
For  example,  to  find  the  value  of  af  when  x  =  0. 

Let  y  =zxi'] 

then  log  2/  =  ic  log  a;  =  0  (—  00  ),  when  a;  =  0. 

Hence,  log^/ =  ^^  =  --,  when  a;  =  0. 

dx^     ^ 
therefore  log  af  =  0,  when  a;  =  0 ;  hence  af  =  1,  when  a?  =  0. 


C-L 


EVALUATION  OF  INDETERMINATE  FORMS.  97 

PROBLEMS. 
Find  the  true  values  of  the  following  functions : 

Ans.  0. 
Ans.  00. 
Ans.  0. 

Ans,  1. 

Ans.  0. 
Ans.  a. 

Ans.  —  1. 

1 
^• 

Ans.  1. 

Ans.  J. 

Ans.  ^. 

Ans.  1. 
Ans.  1. 
u4ns.  e. 
Ans.  e*. 

^ns.  1. 


1. 

loga;^ 

when  a;  =  00  . 

2. 

tanic 
3x' 

when  x  =  ^'n: 

3. 

1  —  \ogx 

when  a;  =  0. 

e' 

. 

4. 

log  tan  2  X 
log  tan  X 

when  a;  =  0. 

5. 

.T"    log  Xy 

when  a;  =  0. 

6. 

2'sin|, 

when  a;  =  00. 

7. 

sec  a;  ic  since  —  -  , 

when  a;  =  ^. 
2 

8. 

2             1 
ar^-1      x-1' 

when  a;  =  1. 

9. 

X             1 

when  a;  =  1. 

log  aj     log  X 

10. 

cosec^  ^  ~  ;;2» 

when  a;  =  0. 

11. 

1 

12. 

2               1 

when  a;  =  0. 

sin^  X     1  —  cos  X 

/IN  tan  X 

when  X  =  0. 

13. 

a;"", 

when  x  =  0. 

14. 

(«"+l)", 

when  a;  =  00. 

15. 

(-?• 

when  a;  =  00. 

16. 

sin  a;**", 

when  a;  =  ^. 

(01- 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


Art.  63.     Compound  Indeterminate  Forms. 

♦  When  a  given  function  can  be  resolved  into  factors,  one  or  more  of 
which  become  indeterminate  for  a  particular  value  of  x,  the  true  value 
may'be  obtained  by  getting  the  true  value  of  each  factor  separately. 

When  the  true  value  of  any  indeterminate  form  is  found,  that  of 
any  constant  power  of  it  can  be  determined. 

PROBLEMS. 

This  may  be  put  in  the  form —  • —j 

J.  ~\~  Qu       J.  —  Xi 

in  which  the  second  factor  only  is  indeterminate. 

<f>{x)      1-x^' 

then  -^-M  =  -  f'\  =  ^af-'=  ^  when  a;  =  1. 

<f>'x       —  bx^-^     b  b 


Therefore  |  '^ — ^^  |  =  — • 


2.  (fllzilt^E!^,  whenc.  =  0. 

(e*  - 1)  tan^  x  ^  /tan  xV  {f  - 1) 

a^  \     X    J  X 

P_^l=l,  andr^l::il]=l; 

L  ^  Jo  L   ^   Jo 

therefore  ^^ — ~    ;  an  a;  _  ^^  \f\^Q^  a;  =  1. 

3.  log(l  +  x  +  a^  +  loga-^+'^^  ^hen  r.  =  0. 

sec  X  —  cos  X 


4. 


(ar^  —  a^  sin  1^  ,   ' 
£^,                when  x  —  a. 

a^cosg^ 

5.    a;"* (sin  a?)  ^'"'f  J  7  ^/Y,  when  a;  =  J. 
\2  sin  2xj  2 


Ans,  1. 

Ans. 

1 
a'' 

Ans. 

2»»+3 

4*f, 


CHAPTER   X. 

MAXIMA  AND   MINIMA   OF   FUNCTIONS. 

Art.  64.     Defixitioxs  ax^d  Geometric  Illustration. 

A  maximum  value  of  a  function  is  a  certain  value  at  which  the 
function  changes  from  an  increasing  to  a  decreasing  function.  Or,  in 
other  words,  f{x)  is  a  maximum  for  that  value  of  x  which  makes  f{x) 
greater  than  f{x  +  li)  and  fix  —  li)  for  very  small  values  of  li. 

A  minimum  value  of  a  function  is  a  certain  value  at  which  the 
function  changes  from  a  decreasing  to  an  increasing  function.  Or, 
f{x)  is  a  minimum  for  that  value  of  x  which  makes  f{x)  less  than 
f{x  +  h)  and  f(x  —  h)  for  very  small  values  of  h. 

In  Fig.  8,  let  the  curve  AB  be  the  locus  of  y=f(x). 

Then  PN  represents  a  maximum  ordinate,  and  P'T  a.  minimum  or- 
dinate.    As  X  increases  toward  OiV,  y  approaches  a  maximum  value, 


Fig.  8. 


PN,  and  the  tangent  to  the  curve  makes  an  acute  angle  with  the 
X-axis.  At  the  point  P,  the  tangent  line  is  parallel  to  the  X-axis. 
Immediately  after  passing  P,  the  tangent  makes  an  obtuse  angle  with 
the  X-axis.     But  by  Art.  27,  the  slope  of  the  tangent  line  is  equal 

99 


K 


100  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

to  f(x) ;  hence  f\x)  is  positive  before,  and  negative  after,  a  maximum 
ordinate.  Likewise  it  may  be  shown  that  f'(x)  is  negative  before,  and 
positive  after,  a  minimum  ordinate.  Thus,  f\x)  =  0,  at  both  maxi- 
mum and  minimum  ordinates.     Therefore,  a  condition  for  both  maxima 

and  minima  is  —  =  0. 
dx 

Art.  65.     Method  of  Determining  Maxima  and  Minima. 

For  a  maximum  value  of  the  function,  f'{x)  =  0  and  f{x)  changes 
sign  from  +  to  —  when  x  passes  through  a  value  corresponding  to  a 
maximum  value  of  the  function.  For  a  minimum  value  of  the  func- 
tion, f\x)  =  0  and  f'{x)  changes  sign  from  —  to  +  when  x  passes 
through  a  value  corresponding  to  a  minimum  value  of  the  function. 
Hence,  the  roots  of  the  equation  f\x)  =  0  are  first  obtained. 

If  a  is  a  root  of  this  equation,  a  value  slightly  less,  and  then-  one 
slightly  greater  than  a,  are  substituted  for  x  in  f\x).  Let  li  repre- 
sent a  very  small  quantity. 

If  /'(a  —  h)  is  +,  and  f(a  +  ^)  is  — ,  then  /(a)  is  a  maximum.   ' 

If  f\a  —  /i)  is  — ,  and  /'(a  +  7i)  is  -}-,  then  /(a)  is  a  minimum. 

For  example,  let    y  =  h-\-{x  —  a)*. 

ay  *-^  ^         ,V.—7A»    "' 

Then  -^  =  4  (aj  —  a)^  =  0 :  heno«  ic  =  a.     "^ 

dx 

Substituting  a  —  h  for  ic,  gives 

dx 
Substituting, a.  +  h  for  x,  gives 

dx 

Here,  f\x)  changes  sign  from  —  to  -h  at  x  =  a\  hence,  a  is  the 
value  of  X  which  gives  a  minimum  function.  Therefore,  y  =  b,  sl  mini- 
mum. 

By  reference  to  Fig.  8,  it  will  be  seen  that  P'^'W  is  a  minimum 
ordinate,  and  the  tangent  to  the  curve  at  this  point  is  perpendicular  to 


MAXIMA   AND  MINIMA   OF   FUNCTIONS.  101 

the  X-axis.  In  this  case  f'{x)  changes  sign  by  passing  through  oo . 
Any  variable  can  change  its  sign  only  by  passing  through  0  or  oo ,  but 
it  does  not  necessarily  follow  that  there  is  a  change  of  sign  whenever 
f'(x)=0,  or  f'(x)=cc  .  At  point  P^^,  the  tangent  is  parallel  to  the 
X-axis,  hence  f'(x)  =  0 ;  but  f'(x)  is  -f  immediately  before  and  after 
reaching  this  value.  Therefore,  the  values  of  x  which  make  /'(x)  =  0 
do  not  always  give  maxima  and  minima,  so  they  are  simply  called 
critical  values,  or  values  for  which  the  function  is  to  be  examined. 

It  is  evident  also  that  a  function  may  have  several  maxima  and 
minima,  and  a  minimum  value  may  be  greater  than  a  maximum  value 
of  the  same  function. 

Art.  66.     Conditions  for  Maxima  and  Minima  by  Taylor's 

Theorem. 

Let /(a;)  have  a  maximum  or  minimum  value  when  0?=  a.   ^ 
Then  if  h  be  a  very  small  increment  of  x,  by  Art.  64, 

f(a)  >f(a  -h  h),  and  f(a)>f(a  —  h),  for  a  maximum, 
also        f(a)  <f(cb  -j-  h),  and  /(a)  <f(a  —  h),  for  a  minimum. 

Therefore        /(a  +  h)  -f(a)  and  f(a  -  h)  -f(a) 
are  each  negative  for  a  maximum,  or  are  each  positive  for  a  minimum. 
Now  by  Taylor's  Theorem, 

f(a  +  h)  -f(a)  =     f'{a)h+f<\a)^  +  f"\a)^+  ....  (1) 

f(a  -  h)  -f(a)  =  _/'(«)  A  +/"(a)|-/'"(a)|+  ....  (2) 

For  a  maximum :  The  first  members  of  (1)  and  (2)  must  be  nega- 
tive, therefore  the  second  members  must  be  negative.  Now  if  h  be 
taken  sufficiently  small,  the  first  term  in  each  second  member  can  be 
made  numerically  greater  th^n  the  sum  of  all  the  terms  following  it; 
hence,  the  sign  of  each  second  member  will  be  the  same  as  that  of  its 
first  term.     But  the  first  terms  have  different  apparent  signs,  so  the 


102  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

second  members  cannot  both  be  negative  unless  the  first  term  disap- 
pears, hence 

/'(a)=0. 

Now  the  first  of  the  remaining  terms  of  the  second  members  con- 
tain h^,  and  these  terms  determine  the  signs  of  the  members.  In 
order  that  these  terms  may  be  negative,  /"(a)  must  be  negative,  or 

/"(a)<0. 

Therefore,  if  f(a)  is  a  maximum, 

f(a)  =  0  and  /"(a)  <  0. 
Similarly,  it  may  be  shown  that  if  /(a)  is  a  minimum, 

/'(a)  =  0  and/"(a)>0. 

However,  if  /"(a)  ^  0,  then  the  sign  of  the  second  members  of  (1) 
and  (2)  will  depend  on  the  terms  containing /'"(a),  and  since  the  terms 
containing /'"(a)  have  opposite  signs,  there  can  be  neither  a  maximum 
nor  a  minimum  unless /"'(a)  also  vanishes;  and  if /'"(a)  =  0,  then 
/(a)  is  a  maximum  when  /^'^(a)  is  negative,  and  a  minimum  when  /^^(a) 
is  positive ;  and  so  on. 

Rule:  Find  f'{x),  and  solve  the  equation,  f'(x)  =  0.  Substitute  the 
roots  of  this  equation  for  x  inf"(x).  Each  value  of  x  ivhich  makes  f"(x) 
negative  will  make  f(x)  a  maximum;  and  each  value  which  makes  f"(x) 
positive  will  make  f(x)  a  minimum. 

However,  if  any  value  of  x  also  makes  f"(x)=  0,  substitute  this  value 
in  the  successive  derivatives  until  one  does  not  reduce  to  0.  If  this  be  of" 
an  odd  order,  the  value  ofx  will  give  neither  a  maximum  nor  a  minimum; 
but  if  it  be  of  an  even  order  and  negative,  f(x)  will  be  a  maximum,  if  of  an 
■eveji  order  and  positive,  f(x)  will  be  a  minimum. 

The  solution  of  problems  in  maxima  and  minima  is  often  simplified 
.by  the  aid  of  the  following  principles : 

^  1.  If  any  value  of  x  makes  af(x)  a  maximum  or  minimum,  a  being 
A  positive  constant,  that  value  will  make/(ic)  a  maximum  or  minimum. 
Hence,  a  constant  factor  may  be  omitted. 

J  2.  If  any  value  of  x  makes  [/(a?)]"  a  maximum  or  minimum,  w  being 
a  positive  constant,  that  value  will  make/(.T)  a  maximum  or  minimum. 


MAXIMA  AND  MINIMA  OF  FUNCTIONS.  103 

Hence,  any  constant  exponent  of  the  function  may  be  omitted;  or  if 
the  function  is  a  radical,  the  radical  sign  may  be  omitted. 

3.  If  any  value  of  x  makes  log /(a;)  a  maximum  or  minimum,  that 
value  will  make  f{x)  a  maximum  or  minimum.  Hence,  to  find  a  maxi- 
mum or  minimum  value  of  the  logarithm  of  a  function,  the  function 
only  need  be  taken. 

4.  If  any  value  of  x  makes  f{x)  a  maximum  or  minimum,  that  value 

will  make  — - —  a  minimum  or  maximum.     Hence,  when  a  function  is 

a  maximum  or  a  minimum,  its  reciprocal  is  a  minimum  or  a  maximum. 

5.  If  any  value  of  x  makes  a  +f{x)  a  maximum  or  minimum,  that 
value  will  make  f{x)  a  maximum  or  minimum.  Hence,  a  constant  term 
may  be  omitted.     • 

Each  of  the  preceding  propositions  may  be  readily  proved.  For 
example,  in  (1),  the  first  derivative  of  af{x)  when  placed  equal  to  zero, 
will  give  an  equation  whose  roots  are  the  same  as  the  roots  of  the 
equation  formed  by  placing  the  first  derivative  of  f{x)  equal  to  zero ; 
hence,  the  critical  values  will  be  the  same  in  both  cases. 


PROBLEMS. 


1.   Find  the  maximum  and  Bainimum  values  of 
2 


ana  mmin 
ar»-3a^-9a;  +  5. 


Let  2/  =  a^-3a^-9a;  +  5; 

then  ^  =  3a^_6a;-9. 

dx 

Placing  the  first  derivative  equal  to  zero,  and  finding  the  roots, 
3a;2_g^_9^Q. 

therefore  a;  =  3  or  —  1. 

The  second  derivative  is  — ^  =  6  a?  —  6. 


104 


DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 


When  x  =  3,  —^  =  12,  and  as  this  value  of  x  makes  the  second  de- 

rivative  positive,  it  corresponds 
to  a  minimum  value  of  the 
function. 

When  x=-l,  ^=-12, 

and  this  result  being  negative 
indicates  a  maximum. 

Substituting  these  values  of 
X  in  the  function,  gives,  when 
a;  =  3,  y  =—  22,  a  minimum, 
and  when  x  =  —l,  y  =  10,  a 
maximum. 

These  results  may  be  illus- 
trated graphically  by  construct- 
ing the  locus  of  the  equation. 

In  Fig.  9  it  will  be  seen  that 

there  is  a   maximum   ordinate 

corresponding   to   the   abscissa 

Fig.  9.  —  1?  and  a  minimum  ordinate 

corresponding  to  the  abscissa  3. 

Remark.    It  will  be  very  instructive  to  construct  the  loci  of  the 

equations  in  the  first  few  examples. 

Examine  the  following  functions  for  Maxima  and  Minima : 

2.  2/  =  ar^  —  Sic^-fSic'  +  l.  Ans.  x=l,  gives  a  Maximum,  2 ; 

x  =  3,  gives  a  Minimum,  —  26, 

3.  2/  =  2  ar'  -  21  a;2  +  36  »  -  20.      Ans.  x  =  1,  gives  Max.,  -  3; 

x  =  6,  gives  Min.,  —  128. 

4.  ?/  =  3  ar^  -  9  a.-^  —  27  a;  -f-  30.        Ans.  x  =  -l,  gives  Max.,  45 ; 

X  =  3,  gives  Min.,  —  51. 

5.  y  =  ^^  ~^)^  Ans.  x=\a,  gives  Min.,  JJ a\ 

(X  —  Z  X 


MAXIMA   AND  MINIMA  OF   FUNCTIONS. 


105 


Ans.  This  function  has  no  real  Max.  or  Min. 


7-   y 


1  -\-  X  tan  X 


8.  y  =  xi. 

9.  y 


sma; 


1  +  tan  X 

10.  y  =  sinx  (1  +  cos  x). 

11.  y::=(x-iy(x-^2)\ 


Ans.  x  =  cos  X,  gives  a  Max. 

A71S.  X  =  e,  gives  Max. 
A71S.  X  =  45°,  gives  Max. 

A71S.  X  =  -7  gives  Max. 

Ans.  ic  =  —  4,  gives  Max. ; 
x=l,  gives  Min. ; 
x  =  —  2,  gives  neither. 


GEOMETRIC    PROBLEMS. 

12.    Determine  the  maximum  rectangle  inscribed  in  a  given  circle. 
Assume  an  inscribed  rectangle  as  in  Fig.  10.     Let  the  diameter 
CB  =  cl,  and  the  side  CD  =  x ;  then 

AC  =  -VilF^li^'. 
Denoting  the  area  by  A,  then  A/  ::^B 

A  =  x^d-  —  x^, 
which  is  to  he  a  maximum. 


By  Art.  66,  2,  the  function  x\d'  -  af)  may 
be  used. 


Fio.  10. 


Put 

Then 

Now 


X^((P    _    r^^^ 


y 

dy 
dx 

^=2d'-12x''  =  2d',  whena;  =  0: 


=  2d'x-4.x''  =  0',  liencea;  =  0,  ora;  =  dV}- 


=  —  4  c?^,  when  x  =  c^VJ. 

Therefore,  x  =  c?  V^,  which  is  the  side  of  an  inscribed  square,  will 
give  the  maximum  rectangle. 


106 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


13.    Find  the  greatest  cylinder  which  can  be  inscribed  in  a  given 

right  cone  with  a  circular  base. 

In   Fig.   11,  let   CH  be  a  cylinder 
inscribed  in  the  cone  OAB. 

Given      AM=  a,  and  0M=  h. 

Let  NC  =  X,  and  NM=  y. 

Denoting  the  volume  by  FJ 
then  F=  ■7r:(?y. 


Fig.  11. 


From    the 

similar 

triangles  AOM 

and  CON, 

h_h-y 
a         X 

,  hence  a 

,=«(.- 

■y)- 

Therefore,    V^=  t-^(^  —  2/)V>  which  is  found  to  be  a  maximum  when 

y  =z^h.     Therefore,  the  altitude  of  the  maximum  inscribed  cylinder  is 
one-third  of  the  altitude  of  the  cone. 

14.  Find  the  maximum  cone  which  can  be  inscribed  in  a  sphere 

whose  radius  is  r. 

In  rig.  12,  let  ADB  and  CDB  be 
the  semicircle  and  triangle  which  gene- 
rate the  sphere  and  inscribed  cone  by 
revolution  about  AB. 

Let  CD  =  x,  CB=^y,  and  F=  the 
volume  of  the  cone ; 

then  V=l7roi^y. 

ar^=03x  CA  =  y{2r-y), 

hence,   V=  ^7ry^(2r  —  y),  which  is  the  function  whose  maximum  is 
required.  Ans.  The  altitude  of  the  Max.  cone  =  |r. 

15.  Determine  the  right  cylinder  of  greatest  convex  surface  that 
can  be  inscribed  in  a  given  sphere. 

If  r  =  the  radius  of  the  sphere,  and  x  =  the  radius  of  the  base  of  the 


MAXIMA   AND  MINIMA   OF   FUNCTIONS.  *       107 


cylinder,  then  the  convex  surface  of  the  cylinder  is  4  ttx Vr*  —  a?.    This 

will  be  a  maximum  when  the  radius  of  the  base  is 

V2 

16.  From  a  given  surface  S,  a  cylindrical  vessel  with  circular  base 
and  open  top  is  to  be  made,  so  as  to  contain  the  greatest  amount. 
To  find  its  dimensions. 

Let  a;=  radius  of  base,  y=  altitude,  and  V=  volume  of  a  cylinder. 

Then  4  F^rfy,  (1) 

and  S  =  -rrx'  +  2  7rxy.  (2) 

Differentiating  (1)  and  (2)  with  respect  to  x : 


From  (1), 

dx                          dx 

hence 

dy^_2y^ 
dx           X 

From  (2), 

0  =  27ra;  +  27ra:^4-2 
dx 

hence 

dy  _      x-{-  y^ 
dx             X 

Hence 

2y          x-\-y 

X                      X 

Therefore  2/  =  a;,  or  the  altitude  =  radius  of  base. 

In  this  example,  (2)  might  have  been  solved  for  y  and  this  value  substituted  in 
(1),  and  the  solution  would  have  been  the  usual  one.  But  the  given  solution  is 
in  this  and  similar  examples  much  shorter. 

17.  What  is  the  length  of  the  axis  of  the  maximum  parabola  which 
can  be  cut  from  a  given  right  circular  cone,  knowing  that  the  area  of  a 
parabola  is  equal  to  two-thirds  of  the  product  of  its  base  and  altitude  ? 

Given  BC  =  a,  and  AB  =  h,  in  Fig.  13, 

Let  CM=x,         then  BM  =  a  —  X, 


and    RS  =  2V(a  —  x)x. 


108 


DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


By  similar  triangles, 


a:h::x'.MN\     .-.  MN=-x. 


Fig.  13. 


Hence,  the  area  of  the  parabola  is 


A  =  --x  \/{a  —  X)  Xy 

which  is  a  maximum  when  x  =  \a. 

18.    What  is  the  altitude  of  the  maximum  rectangle  which  can  be 
inscribed  in  a  given  segment  of  a  parabola  ? 


In  Fig.  14,  let  -BOO  be  the  parabolic  segment  and  AO=h. 
Let  OH=x, 


MAXIMA   AND  MINIMA   OF  FUNCTIONS.  109 

then  MH  =  ^2px. 

Therefore,  area  of  MRSN  =  2  ■V2px(h  —  x), 

which  is  a  maximum  when  a;  =  — 

3 

19.  What  is  the  maximum  cylinder  that  can  be  inscribed  in  an 
oblate  spheroid  whose  semi-axes  are  a  and  b  ? 

Ans.  Radius  of  base  =  J  a  V6 ;  altitude  =  -f  6  V3. 

20.  Find  the  maximum  right  cone  that  can  be  inscribed  in  a  given 
right  cone,  the  vertex  of  the  required  cone  being  at  the  centre  of  the 
base  of  the  given  cone.  Ans.  The  ratio  of  the  altitudes  is  J. 

21.  What  is  the  maximum  isosceles  triangle  which  can  be  inscribed 
in  a  circle  ?  Ans.  An  equilateral  triangle. 

22.  What  is  the  altitude  of  the  cone  of  maximum  convex  surface 
that  can  be  inscribed  in  a  sphere  whose  radius  is  3  ? 

Ans.  Altitude  =  4. 

23.  When  is  the  difference  between  the  sine  and  the  cosine  of  any 
angle  a  maximum  ?  Ans.  When  the  angle  =  135°. 

24.  If  the  strength  of  a  beam  with  rectangular  cross-section  varies 
directly  as  the  breadth,  and  as  the  square  of  the  depth,  what  are  the 
dimensions  of  the  strongest  beam  that  can  be  cut  from  a  round  log 
whose  diameter  is  Z)  ?  Ans.  Depth  =  D  V|. 

25.  A  rectangular  box  with  a  square  base  and  open  at  the  top,  is  to 
be  constructed  to  contain  108  cubic  inches.  What  must  be  its  dimen- 
sions so  as  to  contain  the  least  material  ? 

Ans.  Altitude  =  3  inches ;  side  of  base  =  6  inches. 

26.  What  is  the  altitude  of  the  minimum  cone  that  may  be  circum- 
scribed about  a  sphere  whose  diameter  is  10  ?  Ans.  Altitude  =  20. 

27.  A  person,  being  in  a  boat  3  miles  from  the  nearest  point  of  the 
beach,  wishes  to  reach  in  the  shortest  time  a  place  5  miles  from  that 
point  along  the  shore ;  supposing  he  can  walk  5  miles  an  hour,  but  can 
row  only  at  the  rate  of  4  miles  per  hour,  required  the  place  he  must 
land.  Ans.  One  mile  from  the  place  to  be  reached. 


110  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

28.    Find  the  minimum  value  of  y  when  y  — 


Arts.  2/ =  0.32218. 

.29.    Determine  the  greatest  rectangle  which  can  be  inscribed  in  a 

given  triangle  whose  base  =  2 &  and  altitude  =  a.  Ans.  A  =  \ah, 

30.  A  Norman  Avindow  consists  of  a  rectangle  surmounted  by  a 
semicircle.  Given  the  perimeter,  required  the  height  and  breadth  of 
the  window  when  the  quantity  of  light  admitted  is  a  maximum. 

Ans.  Radius  of  semicircle  =  height  of  rectangle. 

31.  A  privateer  must  pass  between  two  lights  A  and  B  on  opposite 
headlands,  the  distance  between  which  is  c.  The  intensity  of  light  A 
at  a  unit's  distance  is  a,  and  the  intensity  of  light  5  at  a  unit's  dis- 
tance is  h.  At  what  point  must  a  privateer  pass  the  line  joining  the 
lights  so  as  to  be  as  little  in  the  light  as  possible,  assuming  the  princi- 
ple of  optics,  that  the  intensity  of  a  light  at  any  point  is  equal  to  its 
intensity  at  a  unit's  distance  divided  by  the  square  of  the  distance  of 
the  point  from  the  light  ?  a  _       ca^ 

a^  +  h^ 

32.  The  flame  of  a  candle  is  directly  over  the  centre  of  a  circle 
whose  radius  is  5 ;  what  ought  to  be  its  height  above  the  plane  of  the 
circle  so  as  to  illuminate  the  circumference  as  much  as  possible,  sup- 
posing the  intensity  of  the  light  to  vary  directly  as  the  sine  of  the 
angle  under  which  it  strikes  the  illuminated  surface,  and  inversely  as 
the  square  of  its  distance  from  the  same  surface  ? 

Ans.  Height  above  circle  =  5  VJ. 

33.  If  the  total  waste  per  mile  in  an  electric  conductor  is  C^r-\-- 

r 

(r  ohms  resistance  per  mile),  due  to  heat,  interest,  and  depreciation, 
what  is  the  relation  between  C,  r  and  t  when  the  waste  is  a  minimum  ? 

Ans.   Cr  =  t. 

34.  In  the  formula,  A^B  =  p'(^  +  ^^^^^^^  it  is  required  to  find  the 

se^  4-  X*         ' 
value  of  X  that  makes  the  variable  factor  — — — -  a  minimum. 

(a;  -  ly 

Ans.  a;  =  2.2. 


MAXIMA   AND   MINIMA  OF   FUNCTIONS.  Ill 

35.  From  the  differential  equation,  20,000,000  ^  =  -100a;,  find 

clx- 

the  equation  of  the  curve  and  the  maximum  ordinate;  the  first  con- 
stant of  integration  being  found  by  making  ^  =  0  when  x  equals  I, 

dx 

and  the  second  constant  of  integration  being  found  by  making  y  equal 
to  zero  when  x  equals  zero. 

Ans.  y  —■ ( Y  and  Max.  ordinate  = 

^      20,000,000^     2  6    /  600,000 

36.  A  statue  whose  height  is  10  feet  stands  on  a  pedestal  8  feet  in 
height,  which  rests  on  a  level  surface.  At  what  point  on  the  horizontal 
plane  through  the  base  of  the  pedestal  does  the  statue  subtend  the 
greatest  angle  ?  Ans.  12  feet  from  centre  of  base. 

37.  If  -u  denotes  the  velocity  of  a  current,  and  x  the  velocity  of  a 
steamer  through  the  water  against  the  current,  what  will  be  the  speed 
most  economical  in  fuel  if  the  quantity  of  fuel  burnt  is  proportional 
to  a;^  ?  Ans.  x  =  ^v. 


Art.  67.     Maxima  and  Minima  of  Functions  of  Two  Indepen- 
dent Variables. 

Let  f(x,  y)  represent  any  function  of  two  independent  variables. 

When  fix,  y)  >f(x  +  h,  y  +  k), 

for  all  very  small  values  of  h  and  k,  positive  or  negative,  the  function 
has  a  maximum  value. 

When  f{x,  y)  <f{x  +  h,y  +  k), 

for  such  values  of  h  and  k,  the  function  has  a  minimum  value. 

Placing  u  =f(x,  y), 

from  Art.  57, 

f(x  +  h,y^k)-f{x,y)  =  h^^^k^f^ 

Now,  by  argument  similar  to  that  of  Art.  ^Q>,  it  may  be  proved  that 
the  sign  oi  f{x-\-h,  y  •\-k)  —f{x,  y)  will   depend  on  h  ;r-+^"^i  ^"^^ 


112  DIFFEREXTIAL   AXD   INTEGRAL   CALCULUS. 

therefore  will  change  sign  with  li  and  A:;  hence,  a  maximum  or  mini- 
mum value  is  impossible  unless 

dx         dy 
And  since  h  and  k  are  independent, 

|i*  =  0,  and  1^  =  0.  (2) 

ax  dy  ^  ^ 

Substituting  A  =  -—-,  B  =  —^-,  and  0= — -,  in  equation  (1),  srives 
dx^  dxdy  dy^  \  y^  ^ 

fix  J^h,y  +  k)  -fix,  2/)  =  1  {Ah'  4-  2  Bhk  +  Ck^)  +  ... 
In  (3),  the  sign  of  fix  +  h,  y  +  k)  —f(x,  y)  will  depend  on 


^|  +  i?Y+(^0-J5^1, 


and  in  order  that  it  may  retain  the  same  sign  for  all  very  small  values 
of  h  and  A:,  it  is  necessary  that  AC  —  B-  should  be  positive;  for  if 
AC  —  B^  be  negative,  it  will  be  possible,  by  ascribing  some  suitable 
value  to  -  to  make  the  whole  expression  change  its  sign.  Hence  as  a 
condition  for  a  maximum  or  minimum, 

B^<AC.  (4) 

It  is  obvious  from  (4)  that  A  and  C  will  have  the  same  sign,  and 
the  sign  of  (3)  thus  depends  on  A. 

Hence,  for  a  maximum,  A  <  0,  and  C  <  0 ; 

and  for  a  minimum,  ^  >  0,  and  C  >  0.  -^ 

Therefore,  the  conditions  established  are  : 
For  either  a  maximum  or  minimum, 

£^  =  0,  ^  =  0,  and  (^^\^^?1. 
dx      'By       '  \dydxj      ex's/ 


MAXIMA   AND   MINIMA    OF   FUNCTIONS.  113 

Also,  for  a  maximum,     — -  <  0  and  — -  <  0, 
and  for  a  minimum,  t"^ > ^  ^^^  -r-.>^- 


Art.  68.     Conditions  for  Maxima  and  Minima  of  Functions  of 
Three  Independent  Variables. 

By  an  investigation  similar  to  that  of  Art.  67,  the  following  condi- 
tions for  a  maximum  or  minimum  value  of  u  =f(x,  y,  z)  are  established : 
For  either  a  maximum  or  minimum. 


dii 
dx 


'    dy        '    dz        '   \dx dyj       dx^  dy^^ 
\dydzj       dy^  dz^'  \ 


d^u  \^      d\  d\ 


dz  dxj       dz^  da? 

Also,  for  a  maximum,  — -  <  0,  — -  <  0,  and  — -  <  0, 

dxr  62/  dz^ 


->0,  _>0,  and  - 


and  for  a  minimum,  — -  >  0,   — -  >  0,  and  -^^  >0. 


PROBLEMS. 

1.   Find  a  point  so  situated  that  the  sum  of  the  squares  of  its  dis- 
tances from  the  three  vertices  of  a  given  triangle  shall  be  a  minimum."* 

Let  (Xi,  2/1),  (^''2?  2/2)  s-nd.  (%  2/3)  be  the  coordinates  of  the  vertices, 
and  (x,  y)  the  given  point. 
Then, 

[(x  -  x,y  +  (2/  -  2/i)T  +  [{X  -  x,y  4-  (2/  -  y,y2  +  [(^  -  ^^3)^  +  (y-  ys)"] 

is  the  function  to  be  a  minimum,  which  may  be  represented  by  u. 
^  =  2(x-x,)+2(x-x,)+2(x-x,), 

g  =  2(2/-2/0  4-2(^-2/0 +2(2/ -2/3), 

*  See  Byerly's  Diff.  Calc,  p.  236. 


114  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


dxdy 


Making  -^  and  -^  each  equal  to  zero : 
dx  oy 

2{x-x^-\-2(x-  X2)  ■j-2(x-x^)  =  0, 

therefore  a;  =  ^i-±^?-±^. 

3 

2(2/-2/0+2(2/-2/2)+2(2/-2/3)  =  O, 

therefore  v  =  ^i  +  ^2  +  ^3. 

^  3 

^O-52  =  36-0>0, 

^  ==  6  >  0. 

Hence,  w  is  a  minimum  when 

and  the  required  point  is  the  centre  of  gravity  of  the  triangle. 

2.  Find  the  maximum  value  of  afy^  (6  —  x  —  y). 

Ans.  Max.  when  x  =  3,  y  =  2. 

3.  Find  the  maximum  value  of  3  axy  —  a^  —  y^. 

Ans.  Max.  when  x==  a,  y  =  a. 

4.  What  is  the  triangle  of   maximum  perimeter  that  may  be  in- 
scribed in  a  given  circle  ?  ^        ^  .,         ,     ,       , 

Ans.  An  equilateral  triangle. 

5.  Find  the  values  of  a;,  y  and  z,  that  make  oi^-\-y^-\-z^-\-x  — 2z 

—  aw  a  minimum. 

Ans.  »  =  -  2, 2/  =  -  i,  2  =  1. 


MAXIMA  AND  MINIMA  OF  FUNCTIONS. 


115 


6.  What  rectangular  parallelopiped  inscribed  in  a  given  sphere  has 
the  maximum  volume  ?  Ans.  A  cube. 

7.  An  open  vessel  is  to  be  constructed  in  the  form  of  a  rectangular 
parallelopiped,  cax3able  of  containing  108  cubic  inches  of  water.  What 
must  be  its  dimensions  to  require  the  least  material  in  construction  ? 

Ans.  Length  and  width,  6  inches ;  height,  3  inches. 

8.  Prove  that  the  point,  the  sum  of  the  squares  of  whose  distances 
from  n  given  points  situated  in  the  same  plane  shall  be  a  minimum,  is 
the  centre  of  mean  position  of  the  given  points. 


CHAPTER   XI. 

TANGENTS,  NORMALS  AND  ASYMPTOTES   TO   ANY  PLANE  CURVE. 

Art.  69.     Equations  of  the  Tangent  and  Normal. 

Let  y  =f(x)  be  the  equation  of  any  plane  curve  AB,  and  (x',  y')  the 
coordinates  of  the  point  of  tangency  T,  in  Fig.  15.     The  equation  of  a 


Fig.  15. 


straight  line  through  T  is  y  —y'  =  a(x  —  ic'),  in  which  a  is  the  tangent 
of  the  angle  which  the  line  makes  with  the  X-axis. 

dy^ 
dx' 


By  Art.  27, 


tSLJUj/ 


Therefore 


y-y 


dy' 


dx 


■Xx-x')  • 


(1) 


is  the  equation  of  the  tangent  to  the  curve  y  =  f(x)  at  the  poinlf  (x'y  y^).- 

As  the  normal  to  the  curve  at  the  point  T  is  perpendicular  to  the 

dx' 
tangent  at  that  point,  the  slope  of  the  normal  is -,  and  hence  the 

dy' 
equation  of  the  normal  is 


y-y' 


dx' .         ^ 
-(x—x'). 

116 


(2) 


TANGENTS,  NORMALS  AND  ASYMPTOTES. 


117 


Art.  70.  Lengths  of  Tangent,  Normal,  Subtangent  and 

Subnormal. 

In  Fig.  15,  let  TM  be  the  ordinate,  TR  the  tangent  and  TN  the 
normal,  at  the  point  of  contact ;  then  MR  is  the  subtangent  and  MN 
the  subnormal. 

Dir  TM  ,dx' 

RM= =  V — -: 

tan  MRT     ^  dy^' 


hence 


hence 


hence 


hence 


dx' 
subtangent  =  v' 

dy' 

MN=MTtSinMTN=y'^: 

dx' 

subnormal  =  y'—; 
'   dx' 


RT=^V(MRy-\-(MTy  =  -yJ(y'^'+y''; 


tangent  =  y 


=V'^ 


'2    _L.f    yl 


(]?r\. 


dx'J  ' 


TN=V{MTy  +  {MN) 

normal  =  2/'.Jl+^ 


(3) 


(4) 


(5) 


(6) 


Art.  71.  Tangent  of  the  Angle  between  the  Radius  Vector 
AT  Any  Point  of  a  Plane  Curve  and  the  Tangent  to  the 
Curve  at  that  Point,  in  Polar  Coordinates. 

Let  0  be  the  pole,  OX  the  initial  line,  and  P  any  point  of  the  curve 
AB,  in  Fig.  16. 

Let  (r,  0)  be  the  coordinates  of  P,  and  (r  +  Ar,  B  -\-  A^)  be  the 
coordinates  of  another  point  R  of  the  curve.  If  PS  is  perpendicular 
to  OR,  then 

PAS'  =  rsinA^, 

and  SR=(r-\-  Ar)  —  r  cos  Ad. 

rsin  Ad 


Therefore        \>?,TiSRP  = 


r  +  Ar  —  r  cos  Ad 


118 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


Now,  if  the  point  R  approaches  P,  then  the  secant  EP  approaches 
the  tangent  PT',  and  the  angle  /iSi^P  approaches  the  angle  OPT,  If 
the  angle  OPT  is  represented  by  <f>,  then 


Fig.  16. 


tan<^  =  limit 


r  sin  A^ 


r  +  A?'  —  rcosA^ 
r  sin  A^ 


=  limit 


AO 


2_      Ar 

A^  A^ 


mw,  limit^  =  l;     limit  ^  =  ^*; 


2sin2 


limit 


A^ 


limit 


A^  2 


Therefore 


tan  <^  =  r 


d6 
dr 


(1) 


Art.  72.     Derivative  of  an  Arc. 

In  Fig.  17,  let  P  and  P'  be  two  points  on  the  curve  AB  separated 
by  a  short  distance  As.     The  coordinates  of  P  and  P'  are  (x,  y)  and 

(x  4-  AXj  y  +  Ai/)  respectively. 


TANGENTS,   NORMALS  AND   ASYMPTOTES. 
In  the  right  triangle  PMP\ 


hence 


As 
Aa; 


A«-^i+/A?/^^ 


Aic 


Y 

/ 

F 

/ 

Ax 

Ay 

M 

— 

/ 

A 

X 

y 

0 

X 

Fia.  17. 


Now,  when  P'  approaches  P, 


limit  ^  =  limit  Jl  +  W, 
Aa;  ^         \Aa;y 


or 


ds 
dx 


=V'H2)' 


119 


Art.  73.     Derivative  of  an  Arc  in  Polar  Coordinates. 

From  Fig.  16,  regarding  the  limiting  triangle  of  PSR, 

limit  sec  PBS  =  limit  ^  =  limit  — , 
BS  Ar 


hence 


therefore 


sec<^  = 


ds 
dr 


(1) 


I  =  VI  +  tan^ <^  =V^+r^^|J,  by  Art.  71,  (1)  ; 


ds_dsdr 
dO~drde 


120 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Art.  74.    Lengths  of  Tangent,  Normal,  Subtangent,  Subnormal 
AND  Perpendicular  on  Tangent,  in  Polar  Coordinates. 

In  Fig.  18,  let  PT  be  the  tangent  to  the  curve  at  the  point  P,  and 
thrdugh  P  the  normal  PN  is  drawn. 


NT  is  drawn  through  the  pole  0,  perpendicular  to  OP, 
PT  is  called  the  2?o/ar  ton^eni; 
PiV,  the  poZar  iiormal; 
OT,  the  polar  subtangent; 
ON,  the  polar  subnoi'mal; 
and  OD,  the  perpendicular  on  tangent  from  the  pole. 

Or=  subtangent      =  OP  tan  OPT=  Opfr—\  by  Art.  71, 

dr 
0N=  subnormal       =  OP  tan  OPiY=  OP  cot  OPT 

_  dr 


(1) 


(2) 


TANGENTS,  NORMALS  AND  ASYMPTOTES.  121 

PT  =  tangent  =  ^'OW+OT'  =  ry^l  +  '^(j^'  (3) 

PN  =  normal  =  VOW+OW'  =  ^r^  +  f—Y-  (4) 

(5) 


0Z>  =  perpendicular  =  OP  sin  OPD 

tan  OP/;  7^ 


Vl  +  tan-  OPD 


PROBLEMS. 


^-KS)' 


1.  Find  the  equations  of  the  tangent  and  normal  to  the  circle, 

a^  +  2/2  =  r2. 

By  differentiation  ^  =  _?;...  ^'  =  _  ^'. 
dx         y         clx'         y' 

Substituting  this  derivative  in  Art.  69  (1),  gives 

y-y'  =  -  -,{^  -  ^'), 

whence,  xx'  -f-  yy'  =  r^,  which  .is  the  equation  of  the  tangent. 

Substituting  ~^|  =  _  ^|  in  Art.  69  (2),  gives  y  -  y'=  ^^(x  -  x% 
cix  y  X 

which  is  the  equation  of  the  normal. 

2.  Find  the  equations  of  the  tangent  and  normal  to  the  ellipse 

aV  +  b-x'  =  a-b\ 

3.  Find  the  equations  of  the  tangent  and  normal  to  the  parabola 

y^  =  2  px. 

4.  Find  the  equations  of  the  tangent  and  normal  to  y^  =  2a^  —  a:^ 
at  x  =  l.  Ayis.  Equations  of  tangent :  y  =  ^x  -{-  ^,  y  z=  —  :^x  —  ^. 

Equations  of  normal :    y  =  —  2x  +  S,   y  =  2x  —  3. 

5.  What  is  the  inclination  of  the  tangent  to  the  curve 

a^if  =  a\x  +  y),  at  the  origin  ?  Ans.  135°. 

6.  What  is  the  value  of  the  subtangent  to  2/ =  a'' ?  Ans.  m. 


122  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

7.  What  is  the  value  of  the  subnormal  to  i/"  =  a^'^a  ?      Ans.  -^» 

nx 

8.  At  what  angle  do  the  curves  y  =  — — -— ^  and  0^  =  4:  ay  intersect  ? 

Ans.  71°  33' 54". 

9.  Find  the  values  of  the  normal  and  subnormal  to  the  cycloid 

a;  =  2  arc  vers  |  —  V4  y  —  y^^  at  the  point  where  y  =  l. 

Ans.  Normal  =  2 ;  subnormal  =  V3. 

10.  Find  the  values  of  the  tangent,  normal,  subtangent  and  sub- 
normal to  the  spiral  of  Archimedes,  whose  equation  is  r  =  ad. 

Ans,  Subtangent  =  - ,  subnormal  =  a, 
a 


tangent      =  r\l  +  — ,  normal  =  Vr^  +  a*. 
11 .   Find  the  values  of  the  subtangent  and  subnormal  to  r'^  =  a*  cos  2  $. 


Ans.  Subtangent  = — 


a^ sin  2d' 


a 

subnormal  = sin  2  d. 

r 


Art.  76.     Rectilinear  Asymptotes. 

An  asymptote  to  a  curve  is  the  limiting  position  of  the  tangent 
when  the  point  of  contact  moves  to  an  infinite  distance  from  the 
origin. 

Hence,  any  curve  will  have  an  asymptote  when  the  point  of  con- 
tact of  a  tangent  is  infinitely  removed  from  the  origin,  and  when  the 
tangent  intersects  either  coordinate  axis  at  a  finite  distance  from  the 
origin. 

From  Art.  69  (1), 

Intercept  on  X=x'-y'—,  (1) 

dy' 

dv' 
and  intercept  on   F=y'  — a;'-=^-  (2) 

ttX 


TANGENTS,   NORMALS   AND  ASYMPTOTES.  123 

If  in  (1)  and  (2)  the  intercept  on  X  or  F  is  finite  when  a'  =  oo  or 
y'  =  00,  then  the  tangent  at  {x',  y')  is  an  asymptote.  For  example,  to 
examine  the  hyperbola  aV  —  b^^  =  —  a^b'^  for  asymptotes : 

Here  ^'=^'. 

dx'     a^y' 

Hence  intercept  on  X=  x' f-^ 

0  X 

=  -  =  0,  when  x'  =  oo. 
a:' 

Intercept  on  Y=  y' -—  = ^,  =  0,  when  y'  =  oo. 

ay  y' 

Hence,  there  is  an  asymptote  passing  through  the  origin. 

dy'     b'x'         b       1  b      , 

zn  =  -T-t  =  ± ,  =  ±  -»  when  ic  =  oo. 

dx'     cry  a  ^2         a 


X/1- 


x^ 


Therefore,  there  are  two  asymptotes  whose  slopes  are  ±  -,  and 
the  equations  of  the  asymptotes  are  y  =  ±  -  x. 

If,  when  a;'  =  CO  in  (1)  and  (2),  the  intercepts  on  both  X  and  Yare 
infinite,  the  curve  has  no  asymptote  corresponding  to  ic'  =  00. 

If  when  y'  =  cc  in  (1)  and  (2),  the  intercepts  on  X  and  Y  are 
infinite,  the  curve  has  no  asymptote  corresponding  to  ?/'  =  00. 

If  both  intercepts  are  zero,  the  asymptote  passes  through  the  origin, 
and  its  direction  is  found  by  evaluating  -^  for  aj  =  00. 

Art.  76.     Asymptotes  Parallel  to  an  Axis. 

When  a;  =  00  in  the  equation  of  a  curve  gives  a  finite  value  of  y, 
then  there  is  an  asymptote  parallel  to  the  X-axis.  For  instance,  if 
y=  a  when  ic  =  00  in  the  equation  of  the  curve,  then  y  =  ais  the  equa- 
tion of  an  asymptote,  because  it  is  the  equation  of  a  straight  line 
passing  within  a  finite  distance  of  the  origin,  and  touching  the  curve 
at  an  infinite  distance. 

Likewise,  when  ?/  =  00  gives  a;  =  6  in  the  equation  of  a  curve,  then 
a;  =  6  is  an  asymptote  parallel  to  the  F-axis. 


124  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 


For  example,  taking  the  curve  whose  equation  is  ]f-  = 


X  —  b 
Here,  y  =  go    when  a?  =  6 ;  hence  x  —  b  =  0  is  the  equation  of  an 

asymptote  to  the  curve  parallel  to  the  y-axis. 

Art.  77.     Asymptotes  determined  by  Expansion. 

An  asymptote  may  sometimes  be  determined  by  solving  the  equa- 
tion of  the  curve  for  y  and  expanding  the  second  member  into  a  series 
in  descending  powers  of  x. 


For  example,  to  examine  y"^ 


x  —  b 


Here  ^s^x^/'l-^'j  \ 

hence  y=  ±x  (l  — ] 

as  X  approaches  oo,  (1)  approaches 

y=±{^+l}  (2) 

Hence,  as  x  increases,  the  curve  (1)  is  continually  approaching  the 
straight  line  (2),  and  (1)  and  (2)  become  tangent  when  a;  =  (» ;  there- 
fore, y  =  ±(a;-f--j  are  the  equations  of  two  asymptotes  to  the  curve  (1). 

Art.  78.     Asymptotes  in  Polar  Coordinates. 

If  a  polar  curve  has  an  asymptote,  as  the  point  of  contact  is  at  an 
infinite  distance  from  the  pole,  and  as  the  tangent  line  passes  within  a 
finite  distance  from  the  pole,  the  radius  vector  of  the  point  of  contact 
is  parallel  to  the  asymptote,  and  the  subtangent  is  perpendicular  to  the 
asymptote  and  finite.     [See  Fig.  18,  Art.  74.] 

Hence,  for  an  asymptote,  the  polar  subtangent  r^—  is  finite  for 

r  =  00.     Therefore,  to  examine  a  polar  curve  for  an  asymptote,  a  value 
of  0  is  found  which  makes  r  =  x ;  if  the  corresponding  polar  subtan- 


TANGENTS,   NORMALS   AND   ASYMPTOTES. 


125 


gent  is  finite,  there  will  be  an  asymptote,  and  if  the  subtangent  is  in- 
finite, there  is  no  corresponding  asymptote. 

For  example,  to  examine  the  hyperbolic  spiral  rO  =  a  for  asymp- 
totes. 

If  r  =  00  in  r  =  -,  then  ^  =  0. 

e 

f  =  -^,  hence  7-f  =  -a. 
dr         IT  dr 

Therefore  there  is  an  asymptote  parallel  to  the  initial  line  which 
passes  at  a  distance  a  from  the  pole. 


PROBLEMS. 
Examine  the  following  curves  for  asymptotes  : 

Arts.  Asymptote,  yz=  —  x, 

Ans.  No  asymptotes. 


1.  y^  =z  a^  —  Qc^. 

2.  The  circle,  ellipse,  and  parabola. 

3.  2/^  =  ax^  -f-  a^. 


4.  The  cissoid,  y^  = 

5.  f  =  2ax^-x\ 

.3 


2r-x 


a" 


6.  2/  =  c+,        .^2 

(x  —  by 

7.  The  lituus,  rS^  =  a. 


Ans.  y  =  a;-f-. 

Ans.  x  =  2r. 

Ans.  7  =  —  a;  -}-  I  a. 

Ans.  y  =  c,  and  x  =  h. 

Ans.  The  initial  line. 


8.   r  cos  ^  =  a  cos  2  6. 

Ans.  There  is  an  asymptote  perpendicular  to  the  initial  line  at 
a  distance  a  to  the  left  of  the  pole. 


CHAPTER   XII. 


DIRECTION  OF  CURVATURE.     POINTS  OF  INFLECTION. 
OF  CURVATURE.     CONTACT. 


RADIUS 


Art.  79.     Direction  of  Curvature. 

A  curve  is  concave  towards  the  X-axis  at  any  point,  when  in  the 
immediate  vicinity  of  that  point  it  lies  between  the  tangent  and  the 
X-axis.  A  curve  is  convex  towards  the  X-axis  when  the  tangent  lies 
between  the  curve  and  the  axis. 


Fig.  19. 


In  Fig.  19,  let  the  coordinates  oi  P  he  (x',  y').  The  curve  being 
concave  downward,  the  ordinates  of  the  curve  for  the  abscissas  x'  ±  h, 
h  being  a  very  small  quantity,  must  be  less  than  the  corresponding 
ordinates  of  the  line  tangent  to  the  curve  at  P. 

Likewise,  in  Fig.  20,  the  curve  being  convex  downward,  the  ordi- 
nates of  the  curve  for  the  abscissas  x'  ±  h  must  be  greater  than  the 
corresponding  ordinates  of  the  tangent  line  at  P. 

126 


DIRECTION  OF   CURVATURE. 


127 


In  either  case,  let  (x'  +  h,  y")  be  the  coordinates  of  P'. 

If  y  =f(x)  is  the  equation  of  either  curve,  then  y"  =f(x'  +  h),  and 


Fig.  20. 


The  equation  of  the  tangent  at  P  is 

If  the  coordinates  of  Pj  are  (aj'  +  ^,  2/2)? 

Now  2/2=/(^')+/'(^')/^; 

hence  2/"  -  y^  =/"(^')i^  +/'"  (aj')|?  + 


or  if  ^  be  taken  sufficiently  small, 


^ 


A' 


^ 


2/"- 2/.  =/"(»')  I 


(1) 

(2) 
(3) 

(4) 


In  (4),  2/"  — 2/2  will  have  the  same  sign  as /"(a;');  therefore,  the 
curve  is  concave  to  the  X-axis  if  /"(a?')  is  negative,  and  convex  if 
/"  (a;')  is  positive. 

If  the  curve  is  below  the  X-axis,  ?/"  and  y^  are  negative,  and  the 
'Curve  is  convex  towards  the  X-axis  when  —  y"  -{-  y2  is  negative,  that  is, 


128 


DIFFEREl^TIAL   AND  INTEGRAL  CALCULUS. 


iff"{x')  is  negative,  and  tlie  curve  is  concave  towards  the  Jt-axis  when 
/"  (x')  is  positive. 


,Art.  80.     Direction  of  Curvature  in  Polar  Coordinates. 

A  curve  referred  to  polar  coordinates  is  concave  to  the  pole  at  any 
point,  when  in  the  immediate  vicinity  of  that  point  it  lies  between  the 
tangent  to  the  curve  at  that  point  and  the  pole.  A  curve  is  convex 
to  the  pole  when  the  tangent  lies  between  the  curve  and  the  pole. 

If  p  is  the  perpendicular  distance  from  the  pole  to  the  tangent  to 
the  curve  at  a  point  whose  coordinates  are  (r,  ^),  it  is  evident  from 
Fig.  21,  that  when  the  curve  is  concave  to  the  pole,  p  increases  as  r 

increases :  hence,  -^  is  positive. 
dr 


Fig.  21. 


Fig.  22. 


Similarly,  from  Fig.  22,  when  the  curve  is  convex  to  the  pole,  p 

decreases  as  r  increases ;  hence  ^  is  negative. 

dr 

If  the  equation  of  the  curve  is  given  in  terms  of  r  and  By  the  equar 
tion  may  be  transformed  into  an  equation  between  r  and  p  by  aid  of 
Art.  74  (5) ;  then  the  curve  is  concave  or  convex,  according  as  -^  is 
positive  or  negative. 


POINTS   OF   INFLECTION.  129 


Art.  81.     Points  op  InfMction. 

A  point  of  inflection  of  a  curve  is  a  point  where  the  curvature 
changes  from  concavity  to  convexity  or  the  reverse.  Hence,  at  a  point 
of  inflection  the  curve  cuts  the  tangent. 

By  Art.  79,  when  /"  (x)  <  0,  the  curve  is  concave  to  the  X-axis,  and 
convex  when  /"(a;)>0;  therefore  f"(x)  changes  sign,  and  hence 
f"  (x)  =0  or  00  at  a  point  of  inflection. 

For  example,  to  examine  y  = —  for  points  of  inflection. 

d^     6x-2a^^     hence  x  =  ^. 
dx"  b'  '  3 

If  a;<|,  theng<0; 

3  dor 

and  if  x>-,  then  ^,>0. 

3'  dx" 

Hence,  f"{x)  changes  sign  at  the  point  whose  abscissa  is  %  and 
therefore  this  will  be  a  point  of  inflection. 


PROBLEMS. 

1.  Find  the  direction  of  curvature  and  point  of  inflection  of 
y=a+{x  —  bY. 

Ans.  There  is  a  point  of  inflection  at  (b,  a) ;  on  the  left  of  this 
point  the  curve  is  concave,  while  on  the  right  it  is  convex. 

2.  Examine  y  =  x  + 36x^  —  2  x^  —  x^  for  points  of  inflection. 

Arts.  At  x  =  2,  and  x  =  —  3. 

3.  Find  points  of  inflection  and  direction  of  curvature  oi  y  =  — 

X  -\-  xZ 

A71S.  (—  6,  —  I),  (0,  0),  (6,  I);  convex  on  the  left  of  the  first  point, 
concave  between  first  and  second  points,  convex  between  second  and 
third,  and  concave  on  the  right  of  third  point. 

4.  Find  the  direction  of  curvature  of  the  lituus  r  =  — 

Ans.  Concave  towards  the  pole  when  r<aV2; 
convex     towards  the  pole  when  r  >  a^2. 


130 


DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 


Art.  82.     Curvature. 

The  total  curvature  of  a  curve  between  two  points  is  the  total 
change  of  direction  in  passing  from  one  point  to  the  other,  and  is 
measured  by  the  angle  formed  by  the  tangents  to  the  curve  at  the  two 
points. 

The  actual  curvature  of  a  curve  at  a  given  point  is  the  rate  of 
change  of  its  direction  relative  to  that  of  its  length.* 


Fig.  23. 


In  Fig.  23,  let  P  be  any  point  of  the  curve  AB.  The  angle 
PTX=  ip,  or  the  angle  which  the  tangent  at  P  makes  with  the  X-axis 
is  the  direction  of  the  curve  at  P.     Likewise  the  angle  P'SX  is  the 


*  Leibnitz  defined  the  curvature  of  a  curve  at  any  point  as  the  rate  at  which 
the  curve  is  bending,  or  the  rate  at  which  the  tangent  is  revolving  per  unit  length 
of  curve. 


RADIUS   OF   CURVATURE.  131 

direction  of  the  curve  at  P'.     Then  angle  TCS  =  A»/^  is  the  difference 
of  these  inclinations,  and  if  PP'  =  As,  and  the  point  P'  approaches  P, 

limit  ^  =  *^, 
As      cZs 

which  is  an  expression  for  the  curvature. 

If  the  curvature  is  uniform ;  that  is,  if  AB  is  the  arc  of  a  circle 

whose  radius  is  r,  the  angle  TCS  =  angle  PMP'  at  the  centre  subtended 

by  the  arc  PP', 

ppi 
and  angle  TCS  =  arc ; 


r 


hence 


A\f/_ 

_1. 

As 

r 

clip 

1 

ds 

r 

therefore  _r  =  _.  (1) 

ds      r 

Hence,  the  curvature  of  a  circle  is  equal  to  the  reciprocal  of  its 

radius. 

Art.  83.     Radius  of  Curvature. 

The  curvature  of  a  circle  varying  inversely  as  its  radius,  and  as 
any  value  at  pleasure  may  be  given  to  the  radius,  it  follows  that  there 
is  always  a  circle  whose  curvature  is  equal  to  the  curvature  of  any 
curve  at  any  point.  The  circle  tangent  to  a  curve  at  any  point  and 
having  the  same  curvature  as  the  curve  at  that  point  is  called  a  circle 
of  curvature ;  its  centre  is  the  centre  of  curvature,  and  its  radius  is  the 
radius  of  curvature  at  that  point. 

Denoting  the  radius  of  curvature  by  p,  by  Art.  82  (1), 

P=-'  (1) 

Now  it  is  required  to  find  p  in  terms  of  x  and  y. 
By  Art.  27,  tan.//  =  ^-S^; 


hence  sec^  i^r?i// 


dx 
dx  ' 


(^l  d^ 

therefore  ^^=_^  = ^  (2) 

»  ^     secV      n_(^^Y 


132  DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 

Substituting  in  (1)  the  value  of  dif/  just  found  and 
ds 


=  dxyjl  +  f^^]  from  Art.  72, 


\dx 


P  = 


['Ht)'j 


dip 


d^ 
daf 


(3) 


which  is  the  required  radius  of  curvature. 

If  I  represents  the  length  of  the  curve,  equation  (3)  may  be  reduced 

dl^ 

to  the  formula  p  = —. 

dx '  dy 


Art.  84.     Radius  of  Curvature  in  Terms  of  Polar  Coordinates. 

Formula  (3),  Art.  83,  is  first  transformed,  any  quantity  t  being  taken 

as  the  new  independent  variable.     The  values  of  -^  and  —^  from  (2) 

dx  dor 

and  (3)  of  Ex.  3,  Art.  50,  being  substituted  in  (3)  of  Art.  83,  putting 
t  =  0, 

dx^     dy^  ^     d^   dx  _  d^  dy 
d&'      de^         dO^'dO     de'dO 


P  = 


dx^ 
d^ 


da^     df\l 
dO'     dOy 


d^ 


d^y    dx  _  d^x    dy 
d$^"dd~d^'  dO 


(1) 


From  the  equations  of  transformation,  x  =  r  cos  $,  and  y  =  r  sin  0, 
by  differentiation, 

dx  •     /I  .  A  dr 

|  =  -.cos.-2sin.|  +  cos.g, 
g  =  -.sin.  +  2cos.|  +  sin.g. 


CONTACT.  133 

Substituting  these  values  in  (1), 


[^-S] 


(2) 


which  is  the  required  formula. 


Art.  85.     Contact  of  Different  Orders. 

Let  y=f{x)  and  y  =  <f>(x)  be  the  equations  of  any  two  curves  re- 
ferred to  the  same  axes. 

Giving  to  ic  a  small  increment  h,  and  expanding, 

f(x+h)  =  f(x)  +f{x)h+r{x)^+r\x)^  +  ..^,     (1) 

^(xJrh)  =  <^{x)  +  <t>'(x)h  +  i>"(^)  I  +  ct>"'(^)  I  +  -.  (2) 

If  the  two  curves  have  a  common  point  whose  abscissa  is  a,  then 
f(a)  =  <f}  (a).  If,  furthermore,  f'(a)  =  <f>'(d),  the  curves  have  a  com- 
mon tangent ;  this  is  called  contact  of  the  first  order. 

If,  also,  f'\x)  —  <l>"(x)y  the  two  curves  have  contact  of  the  second 
order. 

In  general,  two  curves  will  have  contact  of  the  nth.  order  at  a;  =  a, 
when  the  following  conditions  are  satisfied : 

f(a)  =  <^ (a),  f'(a)  =  <^'(a),  f"(a)  =  cf>"(a),  ...  f%a)  =  <^"(a). 

If  the  curves  have  a  common  point  at  a;  =  a,  and  if  a  be  substituted 
for  X  in  (1)  and  (2),  and  (2)  be  subtracted  from  (1),  then 

/(a  +  h)-<l,{a  +  k)  =  k[f'(a)  -  ,^'(a)]-i'[/"(a)  -  ,^"(a)] 

+  |[/'» -<*"'(»)]  +  -,         (3) 

which  is  the  difference  between  corresponding  ordinates  of  the  curves. 

Now,  if  these  curves  have  contact  of  the  first  order,  the  first  term  of 

the  second  member  of  (3)  reduces  to  zero ;  if  they  have  contact  of  the 

second  order,  the  first  two  terms  reduce  to  zeroj  and  so  on.     Hence, 


134  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

when  the  order  of  contact  is  odd,  the  first  term  which  does  not  reduce 
to  zero  contains  an  even  power  of  h,  and  the  sign  of  the  second  member 
is  the  same  whether  h  be  positive  or  negative ;  therefore,  one  curve  lies 
above  the  other  on  each  side  of  their  common  point,  and  the  curves  do 
not;  intersect.  But  when  the  order  of  contact  is  even,  the  first  term 
which  does  not  vanish  contains  an  odd  power  of  h,  and  in  this  case  the 
second  member  changes  sign  with  h-,  therefore,  one  curve  lies  above 
the  other  on  one  side  of  the  common  point,  and  below  it  on  the  other 
side,  and  the  curves  intersect. 

Art.  86.     Radius  of  the  Osculating  Circle,  and  Coordinates 

OF  its  Centre. 

It  appears  from  Art.  85,  that  w  +  1  equations  must  be  satisfied 
when  a  curve  has  contact  of  the  nth  order  with  another  curve.  As 
an  equation  may  be  made  to  conform  to  as  many  different  conditions 
as  there  are  arbitrary  constants  in  it,  it  follows  that  the  number  denot- 
ing the  order  of  contact  which  any  curve  may  have  is  one  less  than  the 
number  of  arbitrary  constants  in  its  equation.  The  general  equation 
of  the  circle  has  three  constants ;  hence,  at  any  point  of  a  curve,  the 
circle  will  have,  in  general,  contact  of  the  second  order ;  this  circle  is 
called  the  osculating  circle. 

Let  the  equation  of  the  given  curve  be 

2/=/(^),  (1) 

and  the  equation  of  the  circle 

(x'-a)'+(2/'-6)»=r'.  (2) 

Differentiating  (2)  twice, 

a,'_a+(y_6)|.;  =  o,  (3) 

l  +  ©V(/-*)g=0.  (4) 

If  (2)  is  the  osculating  circle  at  the  point  (x,  y)  of  (1), 

x'-x  and  v'-v    ^-%    dy_d^ 
x-x  and2/-y,    ^-^^    dx^' d^ 


CONTACT.  135 

Substituting  these  values  in  (4),  and  solving  for  h : 


'=y+-^-  <^> 


From  (3),  a  =  x 


dx 


1^,'dy 


dy 
dx 


d^ 
da^ 


(6) 


Substituting  values  of  (y  —  h)  and  (x  —  a)  from  (5)  and  (6)  in  (2), 
after  reducing, 


.tM. 


dx- 

The  values  of  a  and  h  in  (5)  and  (6)  are  the  coordinates  of  the 
centre  of  the  osculating  circle,  and  the  value  of  r  in  (7)  is  its  radius. 
Hence,  by  comparison  with  Art.  83,  it  will  be  seen  that  the  osculating 
circle  is  the  circle  of  curvature  and  the  radius  of  the  osculating  circle 
is  the  radius  of  curvature. 

Art.  87.  The  Osculating  Circle  has  Contact  of  the  Third 
Order  where  the  Radius  op  Curvature  is  a  Maximum  or 
Minimum. 

If  p  is  to  be  a  maximum  or  minimum,  by  Art.  %Q, 

^  =  0. 
dx 

Differentiating  (3)  of  Art.  83, 

e^p      2^        dx^J  dx\dx^J      doi^\        dx^J  _^  . 

dx^ jm  '      ^' 

\dx'J 

therefore  d?y        dce[^  ^2) 

dx^       14.^ 
dx" 


136  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

The  same  value  of  — ^  as  found  in  (2)  will  be  obtained  by  differen- 
t^ating  (5)  of  Art.  S6.     Therefore,  the  given  curve  and  the  osculating 

circle  have  the  same  value  for  -^  at  a  point  of  maximum  or  minimum 
,  dxr 

curvature ;  hence  the  contact  at  such  a  point  is  of  the  third  order. 


PROBLEMS. 


1.   Find  the  radius  of  curvature  of  the  parabola  y^  —  2px  at  any 
point,  and  the  coordinates  of  the  centre  of  curvature. 

dy__p      d^y  _      p^. 
dx     y       dx^         y^ 


hence  p  =  ±-^ ^  -         ,       -    , 


r  p'        p'' 


^      ~Sx+p. 


1  +  "' 


/ 


_f       p^ 


At  the  vertex,  a;  =  0  and  2/  =  0 ;  therefore  /o  =  p,  a  =  p  and  &  =  0. 

2.  Find  the  radius  of  curvature  of  the  ellipse  aV  4-  ^^y?  =  a}h^. 

3.  Find  the  curvature  of  2/  =  a;*  —  4  ar'  —  18  a^  at  the  origin. 

Ans.  p  =  ^V 

4.  Find  the   radius  of  curvature   of   the   cycloid   a?  =  r  arc  vers  ^ 

r 


-V2?-2/-/.  Ans.  p  =  2V2ry. 


PROBLEMS.  137 

X  X 

5.  Find  the  radius  of  curvature  of  the  catenary  y  =  -(e'*-{-e  "). 

Arts,  p  =  — . 

6.  Find  the  radius  of  curvature  of  the  spiral  of  Archimedes,  r  =  a9. 

2or  -{-'T 

7.  Find  the  radius  of  curvature  of  the  cardiod  r  =  a  (1  —  cos  B). 

Ans.  p  =  f  V2  ar. 

8.  Find  the  radius  of  curvature  of  the  ellipse  whose  axes  are  8  and 
4,  at  a;  =  2,  and  the  coordinates  of  the  centre  of  curvature. 

Ans.  p  =  5.86 ;  a  =  .38,  and  6  =  -  3.9. 

9.  Find  the  order  of  contact  of 

x^  —  3x-  =  9y  —  27  and  3  x  =  2S  —  9  y.     Ans.  Second  order. 

10.  What  is  the  order  of  contact  of  the  parabola  Ay  =  a^  —  4:  and 
the  circle  x^  -\-y^  —  2y  =  3?  Ans.  Third  order. 

11.  What  is  the  radius  of  curvature  of  the  curve  16  ?/^  =  4  a;*  —  a^, 
at  the  points  (0,  0)  and  (2,  0)  ?  .  Ans.  p  =  1,  and  p  =  2. 

12.  What  is  the  radius  of  curvature  of  the  curve  y  =  oi^-\-5oiy^-\-6x, 
at  the  origin  ?  Ans.  p  =  22.506. 

13.  Find  the  radius  oF  cur vaturF  and  the  coordinates  of  the  centre 
of  curvature  of  the  curve  ?/  =  e*,  at  a;  =  0. 

Ans.  p  =  2V2,  (a,  b)  =  (-  2,  3). 


CHAPTER  XIIL 


EVOLUTES  AND  INVOLUTES.     ENVELOPES. 
Art.  88.     Definition  of  Evolute  and  Involute. 

The  evolute  of  any  curve  is  the  curve  which  is  the  locus  of  the 
centres  of  all  the  osculating  circles  of  the  given  curve ;  the  given  curve 
with  respect  to  its  evolute  being  called  an  involute. 

^ .P^ 

A 


Fig.  24. 

In  Fig.  24,  let  AB  be  the  given  curve,  and  the  centres  of  curvature 
of  P',  P",  P'",  etc.,  be  respectively  Pj,  Pj,  Pg,  etc.;  then  the  curve  MN, 
which  is  the  locus  of  Pj,  Pj,  P3,  etc.,  is  the  evolute  of  AB.. 

Art.  89.     Equation  of  the  Evolute. 

The  equation  of  the  evolute  is  the  equation  which  expresses  the 
relation  between  the  coordinates  of  the  centres  of  all  the  osculating 

138 


EVOLUTES  AND  INVOLUTES. 


139 


circles  of  the  involute.     The  values  of  -^  and  —.  derived  from  the 

dx  dar 

equation  of  the  curve  are  substituted  in  equations  (5)  and  (6)  of  Art. 
86,  giving  two  equations,  which,  together  with  the  equation  of  the  given 
curve,  make  three  equations  involving  ic,  y,  a  and  b ;  by  combining  these 
equations,  eliminating  x  and  y,  a  resulting  equation  will  be  obtained 
showing  a  relation  between  a  and  b,  the  coordinates  of  the  evolute, 
which  is  the  required  equation. 


Fig.  25. 


For  example,  to  find  the  equation  of  the  evolute  to  the  common 
parabola,  y^  =  2px. 

Here  ^^l^^^^l. 


dx     y   da^ 

Substituting  in  (5)  and  (6)  of  Art.  S6:      ' 

6^3,-^.  ^  =  -i!!i  hence  2/»  =  f,*6i. 


P 


y       y  p"  3 


(2) 
(3) 


The  values  of  y^  and  x  in  (2)  and  (3)  substituted  in  the  equation  of 
the  parabola,  give  p^b^  =  2p^  ~^; 


therefore 


^'=i-/^-py 


(4) 


140  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Equation  (4). is  the  equation  of  the  evolute. 
This  evolute  is  called  the  semi-cubical  parabola. 
Constructing  the  evolute,  its  form  and  position  is  as  shown  in  Fig. 
25,  where  OA  =  p. 

Q 

•  If  the  origin  is  transferred  to  A,  the  equation  becomes  b^  = a^ 

27p 

or  as  a  and  b  are  the  variable  coordinates,  the  equation  may  be  written, 

-       ^       27p 

The  semi-cubical  parabola  is  so  called  from  the  nature  of  its  equation ;  the 
equation  being  solved  for  y  gives  the  function  expressed  in  terms  of  the  variable 
with  an  exponent  of  three  halves. 


Art.  90.     A  Normal  to  Any  Involute  is  Tangent  to  its 

Evolute. 

Let  P',  in  Fig.  24,  be  any  point  of  the  involute,  whose  coordinates 
are  x'  and  y',  and  let  (a,  b)  be  the  coordinates  of  P^  the  centre  of  curva- 
ture.    Then  the  equation  of  the  normal  at  P'  by  Art.  69  (2),  is 

y-y'  =  -'^,(^-^')-  (1) 

As  (1)  passes  through  (a,  b), 

x'-a+^,(y'-b)  =  0.  (2) 

Now  if  P'  moves  along  the  curve,  Pj  moves  along  the  evolute ;  hence 
a,  b  and  y'  are  functions  of  x'. 
Differentiating  (2), 

dx'     dx'' ^^        ^  ^ [dx'J      dx'  dx'  ^  ^ 

But  since  (a,  b)  is  on  the  evolute,  by  Art.  86  (5), 


6  =  2/'  + 


d'y' 
dx"' 


(.'-6)g+i+i:;=o.  (4) 


EVOLUTES  AND  INVOLUTES.  141 

Substituting  (4)  in  (3), 

•      _da  _  dj/_  db_  _  a  .        _  ^  _  ^, 
dx'     dx'  dx'        '  dy'      da 

Hence,  equation  (2),  which  is  the  equation  of  the  normal  to  the 
involute  at  (x',  y'),  may  be  written 

which  is  the  equation  of  a  tangent  to  the  e volute  at  the  point  (a,  b). 


Art.  91.     The  Difference  between  Any  Two  Radii  op  Curvature 

OF  AN  Involute. 

The  equation  of  the  circle  of  curvature  at  (x',  y')  is 

(x'-ay-{-(y'-by  =  p\  (1) 

Differentiating  (1),  y',  a,  b  and  p  being  functions  of  x',  gives 

(.._.)_(.._.)g+(y_,)|;_(,._,)g=,g.     (2) 

By  Art.  90  (2),  x' -  a -\-^,iy' -b)=0;  (3) 

and  by  Art.  90  (5),  y'-b  =  ^(x^-  a).  (4) 

Combining  (1)  and  the  square  of  (4), 

(.-'—y{^^^=p'-  (5) 

Combining  (2)  and  (3),  and  the  resulting  equation  with  (4), 

^  \     da'      J      da  ^  ^ 

Dividing  (6)  by  the  square  root  of  (5),  and  simplifying, 

■y/da'  +  db^  =  dp. 
Hence,  if  s  represents  the  length  of  the  evolute,  by  Art.  72, 

ds  =  dp', 
therefore  As  =  Ap, 


142  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

or  in  Fig.  24,  P,P^  =  r'P^  -  P'P^, 

or  the  difference  between  any  two  radii  of  curvature  of  an  involute  is 
^qual  to  the  included  arc  of  the  evolute 


Art.  92.     Mechanical  Construction  of  an  Involute  from  its 

E  VOLUTE. 

From  the  two  properties  of  the  evolute  established  in  Arts.  90  and 
91,  the  involute  may  be  readily  constructed  from  its-  evolute.  Thus  in 
Fig.  26,  if  one  end  of  a  string  be  fastened  at  N  and  the  string  be 
stretched  along  the  curve  NM  having  a  pencil  attached  to  the  other 


Fig.  26. 

end,  and  then  the  string  be  gradually  unwound  from  the  evolute, 
always  being  in  tension,  the  pencil  will  describe  the  involute  MA. 
Every  point  in  the  string  beyond  TV- will  describe  an  involute,  as  i? 
describes  RS.  So  while  any  curve  can  have  but  one  evolute,  as  NM 
is  the  only  evolute  of  MA,  it  is  evident  that  any  curve  may  have  an 
infinite  number  of  involutes.  A  series  of  curves  having  the  same 
evolute  are  called  parallel  curves. 

Art.  93.     Envelopes  of  Curves. 
If  in  the  equation  of  a  plane  curve  of  the  form 
f(x,  y,  a)  =  0, 


7^ 


ENVELOPES. 


143 


different  values  be  successively  assigned  to  a,  the  several  equations  thus 
obtained  will  represent  distinct  curves,  differing  from  each  other  in 
form  and  position,  but  belonging  to  the  same  class,  or  family  of  curves. 

Now,  if  a  is  supposed  to  vary  by  infinitesimal  increments,  any  two 
adjacent  curves  of  the  series  will,  in  general,  intersect,  and  the  inter- 
sections are  points  of  the  envelope. 

Hence,  an  envelope  of  a  series  of  curves  is  the  locus  of  the  ultimate 
intersections  of  the  consecutive  curves. 

The  quantity  a,  which  remains  constant  in  any  one  curve,  is  called 
the  variable  parameter. 

?, _P2    P., 


Fig.  27. 

In  Fig.  27,  let  AA\  BB',  etc.,  represent  curves  of  a  series,  and  a„ 
02,  etc.,  their  respective  parameters ;  then  if  ag  —  cii,  as  —  ag,  etc., 
diminish  indefinitely,  the  ultimate  intersections  Pi,  P2,  P3,  etc.,  will  be 
points  of  the  envelope.  And,  at  the  limit,  the  line  Pj,  Pg,  joins  two 
consecutive  points  on  the  envelope  and  on  the  curve  BB',  and  hence  is 
tangent  to  both  the  envelope  and  the  curve  BB',  then  the  envelope  is 
tangent  to  the  curve  BB'. 

Similarly,  it  may  be  shown  that  the  envelope  is  tangent  to  any 
other  curve  of  the  series. 

Hence  the  envelope  of  a  family  of  curves  is  tangent  to  each  curve 
of  the  series. 


Art.  94.     Equation  of  the  Envelope  of  a  Family  of  Curves. 
Let  the  equations  of  two  curves  of  the  series  be 

f(x,y,a)  =  0,  (1) 

and  f(x,  y,a  +  Aa)  =  0.  (2) 


144  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

The  coordinates  of  the  point  of  intersection  of  (1)  and  (2)  will 
satisfy  both  (1)  and  (2),  and  hence  will  also  satisfy 

fix,  y,a^  Aa)  -f(x,  y,  a)  =  0, 

and  f(x,y,a-\-Aa)-f(x,y,a)^^  ,3. 

Aa  ^  ^ 

As  Aa  approaches  0,  the  limit  in  equation  (3)  is 

dfjx,  y,  a)  ^  ^  , ,  V 

da  '  ^  ^ 

Now  the  coordinates  of  the  point  of  intersection  of  two  consecutive 
curves  satisfy  both  (4)  and  (1).  Therefore,  by  eliminating  a  between 
(1)  and  (4)  the  resulting  equation  is  the  equation  of  the  locus  of  the 
ultimate  intersections,  which  is  the  required  equation  of  the  envelope. 

For  example,  required  the  envelope  of  a  series  of  curves  repre- 
sented by 

y  =  ax-l^!^.  (1) 

a  being  the  variable  parameter. 
'    Differentiating  (1)  with  respect  to  a, 

x-^  =  0;  (2) 

2 
hence  '      a  =  —  (3) 

X 


Combining  (1)  and  (2),  eliminating  a,  and  reducing, 

y  =  l- 
which  is  the  equation  of  the  envelope 


2/  =  l-J  (4) 


PROBLEMS. 
1.   Find  the  equation  of  the  evolute  of  the  ellipse 

AY -h  B'x' =  A'B\  (1) 

Here  dy___mo         .   ^__^. 


PROBLEMS.  145 

hence  a  =  ^ -^^,    and      x  =  (^^— ^j  ; 

,^_{A^-B^f  ^^,  (_m\\ 

Substituting  these  values  of  x  and  y  in  (1), 

(Aa)^  4-  {Bbf^  =  {A^  -  JB^)^, 
which  is  the  equation  of  the  required  evolute. 

2.  Find  the  equation  of  the  evolute  of  the  cycloid, 

x  =  r  vers~^  ^  ""  V2  ry  —  y\ 

Ans.  a=r vers"' ( j  +  V—  2 r6  —  6'. 

3.  Find  the  equation  of  the  evolute  to  the  hypocycloid, 

x^  -\-y^  =  A^. 

Ans.  (a  4-  b)^  +(a-b)^  =  2  A^. 

4.  Find  the  envelope  of  y^  -\-(x  —  of  =  16,  in  which  a  is  a  variable 
parameter.  Ans.  y  =  ±  4:. 

5.  Find  the  envelope  oi  y  =  ax  -\-  —,  a  being  the  variable  parameter. 

a 

Ans.  y^  =  4:  mx. 

6.  A  straight  line  of  given  length  slides  down  between  rectangular 
axes ;  required  the  envelope  of  the  moving  straight  line. 

If  c  represents  the  length  of  the  line  and  a  and  b  the  intercepts,  the 
equation  is 

M=^'  ,    (^) 

the  relation  between  a  and  b  being 

a'+b'  =  (^.  (2) 

Differentiating  (1)  and  (2)  with  respect  to  a  and  &,  gives 

^EcUi  =  ^db,  (3) 

a"  b'     '  ^  ^ 

and  —  ada  =  bdb.  (4) 


146  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

Dividing  (3)  by  (4), 

X  y       X     y 

a  b       a     b       1 


heace 
and 


X  _y  _     _  _     ^ 

^~F'  ^^'  '^'~b'~¥TT'~?' 

a  =  (xc^^, 

b  =  (yc^^, 


which,  substituted  in  (2),  gives 

x^  -\-y^  =  c^, 
which  is  the  equation  of  the  hypocycloid. 
Y 


Fig.  28. 


7.  Find  the  envelope  of  a  series  of  concentric  ellipses,  the  area  and 
direction  of  axes  being  constant. 

A71S.  If  c  =  area,  the  equation  of  the  envelope  is  xy  =  ±  ——. 

8.  Find   the  envelope  of  a;  cos  a  +  1/ sin  a  =  p,  in  which  a  is  the 
variable  parameter.  Ans.  ^  -^y^  =  p^. 


CHAPTER  XIV. 
SINGULAR  POINTS. 

Art.  95.     Definitions. 

A  singular  point  is  a  point  of  a  curve  which  has  some  peculiarity 
not  common  to  other  points  of  the  curve,  and  not  depending  on  the 
position  of  the  coordinate  axes. 

The  most  important  singular  points  are  : 

1st.  Points  of  maximum  and  minimum  ordinates;  2d.  Points  of 
inflection;  3d.,  Multiple  points;  4th.  Cusps;  5th.  Conjugate  points; 
6th.  Stop  points ;    7th.  Shooting  points. 

Points  of  maximum  and  minimum  ordinates  have  been  considered 
in  Chapter  X.,  and  points  of  inflection  in  Art.  81. 

Art.  96.     Multiple  Points. 

A  multiple  pfohit  is  a  point  common  to  two  or  more  branches  of  a 
curve. 

There  are  two  species  of  multiple  points :  1st.  Points  of  multi- 
ple intersection,  or  where  two  or  more  branchies  of  a  curve  intersect ; 
2d.  Points  of  osculation,  or  where  two  or  more  branches  are  tangent 
to  each  other. 

Multiple   points   are   double,   triple,    etc.,   as   two,  three,  or   more 
branches  meet  at  the  same  point. 
■     At  a  multiple  point  there  will  be  as  many  tangents,  and  therefore 

as  many  values  of  -^  as  there  are  branches.     If  the  values  of  —  are 
dx  dx 

unequal,  the  multiple  point  will  be  one  of  the  first  species,  but  if  the 

values  of  -^  are  equal,  it  will  be  one  of  the  second  species, 
(tx 

147 


148  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

Let  u  =f{x,  y)  =  0 


(1) 


be  the  equation  of  the  curve  freed  of  radicals. 
'Then,  by  Art.  47, 


dy 
dx 


du 
dx 
du 
dy 


And  since  differentiation  never  introduces  radicals  when  the  func- 
tion contains  none,  the  value  of  -^  cannot  contain  radicals,  and  there- 

dx  ^ 

fore  cannot  have  more  than  one  value  unless  it  assumes  the  form  x' 
Hence  the  condition  for  a  multiple  point  is  ^  =  - . 

Therefore,  to  examine  for  multiple  points,  —  and  —  as  obtained 

dx  dy 

from  the  equation  of  the  curve  are  placed  equal  to  zero,  and  the  corre- 
sponding values  of  «  and  y  are  found.     If  these  values  of  x  and  y  are 

real  and  satisfy  (1),  they  may  determine  multiple  points.     Then  -^=  - 

(XX       u 

is  evaluated  for  the  critical   values  of  x  and  y,  and  every  real  value 
determines  one  branch  passing  through  the  multiple  point. 


PROBLEMS. 
1.    Examine  the  curve  y^  —  {x  —  ayx  =  0  for  a  multiple  point. 
du 


Here  —  =  —  2  (a;  —  a)  a;  —  (a;  —  a)^  =  0 ; 

dx  V  /  V  /  > 

and  ^  =  2  y  ==  0. 

dy 

Solving  (1)  and  (2)  for  x  and  y,  gives 

,  and 


(1) 

(2) 


x  =  a 
2/  =  0 


-I 


But  only  the  first  point  is  to  be  examined,  as  the  second  point  does 
not  satisfy  the  equation  of  the  curve. 


SINGULAR  POINTS. 


149 


dy  _      —2(x  —  a)x—(x  —  ay 3a;  —  a  » 

dx"  2y  ~       2Vx 

=  ±  Va,  when  x  =  a. 

Therefore  the  multiple  point  is  a  double  point  of  the  first  kind,  as 
f       shown  in  Fig.  29. 


Fig.  29. 


2.    Examine  the  curve  x*  -\-2  ax^y  —  ay^  =  0,  for  multiple  points. 

dx 

—  =  2ax'-Say'  =  0. 
.  dy 


(1) 
(2) 


Combining  (1)  and  (2)  gives  three  pairs  of  values  for  x  and  y,  but 
the  only  pair  that  satisfies  the  equation  of  the  curve  is  (0,  0). 

dy^  4:a^-\-4.axy  ^0  ^^en  I  ^  ""  ^ 
dx     3ay^-2ax'     O'  Xy=0. 


dy 


dp 


Evaluating  by  Art.  59,  and  representing  ^  by  p  and  -f-  by  p\ 

cix  dx 


dy  _    _  12  7^  -\-  4:  ay  -{-  4:  axp 
dx  6  ayp  —  4  aa; 


0      ,        {x  =  0 
-,  when  < 

0'  (y  =  o, 


Sap 


,  when 


^x  =  0 


and 


_  24  a;  +  8  aj9  4-  4  axp'  _ 

6  ap'^ -\- 6  ayp' —  4:  a      6ap^  — 4  a 

Hence  p(6  ajs^  —  4  a)  =  8  op ; 

p  =  ^  =  0,  +V2,  and  -V2. 
dx 

Therefore  there  is  a  triple  point  of  the  first  kind  at  the  origin. 


150  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

3.  Examine  y^  —  aV  —  x^  for  a  multiple  point. 

Ans.   There  is  a  double  point  of  the  first  kind  at  the  origin  where 

ax 

4.  Show  that  the  curve  ]^  —  x*  -\-x^  has  a  point  of  osculation  at  the 
origin. 

Art.  97.     Cusps. 

A  (Msp  is  a  point  at  which  two  branches  of  a  curve  are  tangent  to 
each  other  and  terminate. 

Cusps  are  therefore  multiple  points  of  the  second  species. 

There  are  two  kinds  of  cusps :  1st.  When  the  two  branches  lie  on 
-opposite  sides  of  the  common  tangent;  2d.  When  the  two  branches 
are  on  the  same  side  of  the  common  tangent. 

Since  a  cusp  is  a  particular  kind  of  multiple  point,  curves  are 
examined  for  cusps  as  for  multiple  points.  But  as  a  cusp  is  dis- 
tinguished from  a  multiple  point  by  both  branches  stopping  at  the 
point,  the  curve  must  be  traced  in  the  vicinity  of  the  point  in  question 

to  determine  a  cusp.     If  the  two  values  of  J  at  the  cusp  have  con- 
dor 

trary  signs,  the  cusp  is  of  the  first  kind,  and  if  they  have  the  same 

sign,  the  cusp  is  of  the  second  species. 

The  vertex  of  the  semi-cubical  parabola  is  a  cusp  of  the  first  kind. 
[See  point  A,  Fig.  25.] 

The  curve  (?/  —  ^Y  =  ar^  has  a  cusp  of  the  second  species,  determined 
as  follows : 

Taking  the  square  root  of  each  meinber  of  the  equation, 

y^x^±x^',  (1) 

lience  ^  =  2x±\x^,  (2) 

ax 

and  S=2±i^x^.  (3) 

aar 

In  (1),  if  X  =  0,  then  2/  =  0 ;  if  x  is  negative,  y  is  imaginary ;  if  x  is 
positive,  y  has  two  real  values.     Hence,  the  curve  has  two  branches  on 


/  0 


SINGULAR  POINTS. 


151 


the  right  of  the  T^axis  which  meet  and  terminate  at  the  origin.    The 
locus  of  the  equation  is  shown  in  Fig.  30. 


In  (2)   ^  =  0,  when  a;  =  0;    hence  the  X-axis  is  tangent  to  both 
dx 

branches,  and  there  is  a  cusp  at  the  origin. 

In  (3),  when  a  value  slightly  greater  than  0  is  substituted  for  cc,  the 
two  values  of  ^  are  both  positive ;  hence  the  cusp  is  of  the  second 

species. 


t.   Conjugate  Points.     Stop  Points.     Shooting  Points. 


A  conjugate  or  isolated  point  is  a  point  whose  coordinates  satisfy 
the  equation  of  a  curve,  but  through  which  the  curve  does  not  pass. 
As  the  conjugate  point  is  detached  from  the"  curve,  if  the  substitutions 
of  a  +  6  and  a  —  h  for  x  in  the  equation  of  the  curve,  h  being  very 
small,  give  imaginary  values  for  y,  then  there  is  a  conjugate  point 
whose  abscissa  is  a. 

Or,  if  at  any  point  whose  coordinates  satisfy  the  equation  of  a 
curve,  ^  is  imaginary,  this  point  will  be  a  point  through  which  no 

branches  pass,  and  hence  will  be  a  conjugate  point. 

For  example,  to  examine  y^  =  (x  —  ly  (x  —  2)  for  conjugate  points. 

The  point  (1,  0)  will  be  such  a  point,  for  if  some  value  a  little 
greater  or  a  little  less  than  1  be  substituted  for  x  in  the  equation,  the 


152 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


resulting  value  of  y  will  be  imaginary,  yet  the  point  (1,  0)  satisfies  the 
equation.     Or  by  the  second  method : 


3aj-5 


•  ^_ 


N"ow  the  point  (1,  0)  which  satisfies  the  equation  of  the  curve 
makes  —  imaginary,  and  hence  is  a  conjugate  point. 

In  Fig.  31,  JOT"  is  the  curve  and  P  is  the  conjugate  point. 

A  stop  point  is  a  point  of  a  curve  at  which  a  branch  suddenly  ends. 
For  example,  to  examine  y  =  x  log  x  for  a  stop  point.  Here,  for  any 
positive  value  of  x,  y  has  one  real  value ;  when  .t  =  0,  ?/  =  0 ;  when  x  is 
negative,  y  is  imaginary ;  therefore  the  origin  is  a  stop  point. 


Fig.  31. 


A  shooti7ig  point  is  a  point  of  a  curve  at  which  two  or  more  branches 
terminate  without  having  a  common  tangent. 


For  example,  to  examine  y  =  x  tan"^  -  for  shooting  points. 

X 


Here, 


^  =  tan-'l- 
dx 


1+3? 


When  a;  =  0,  then  v  =  0.  and  ^  =  ±  -• 

dx         2 


SINGULAR   POINTS.  153 

If  X  be  positive  and  approach  zero  as  its  limit,  ultimately  ?/  =  0  and 
-^  =  ^;    but  if  X  be  negative,  ultimately  ^  =  0  and  -^  =  —  ^.     Hence 

(XX       Ld  (XX  ^ 

two  branches  meet  at  the  origin,  one  inclined  tan  W  -  j  and  the  other 

inclined  tan^(  —  -Y     Therefore  the  origin  is  a  shooting  point. 

Stop  points  and  shooting  points  occur  only  in  transcendental  curves, 
and  may  be  discovered  in  any  curve  by  tracing  the  curve  in  the  vicin- 
ity of  the  singular  points. 


•  :  CHAPTER  XV. 

INTEGRATION  OF  RATIONAL  FRACTIONS. 

Art.  99.     Rational  Fractions. 

A  rational  fraction  is  one  whose  numerator  and  denominator  are 
rational.  If  the  degree  of  the  numerator  is  equal  to  or  greater  than 
the  degree  of  the  denominator,  the  fraction  can  be  reduced  by  division 
to  the  sum  of  several  integral  terms  and  a  fraction  whose  numerator  is 
of  a  lower  degree  than  its  denominator.     For  example, 

^'  +  ^  ^     dx^x'dx-^-  xdx  -3dx  +   ,  ^^  +  ^     dx, 


x'-\-2x  +1  a^-t-2a;  +  l 

in  which  the  last  term  is  the  only  fractional  term.  So  it  is  necessary 
to  consider  only  rational  fractions  in  which  the  degree  of  the  numera- 
tor is  less  than  the  degree  of  the  denominator. 

A  rational  fraction  is  integrated  by  decomposing  it  into  a  number  of 
simpler  partial  fractions,  which  can  be  integrated  separately. 

Case  1.     When  the  denominator  can  be  resolved  into  n  real  and 

unequal  factors  of  the  first  degree. 

f(x) 
Let  ^^-^  dx  represent  a  rational  fraction,  whose  denominator  may 

be  resolved  into  the  factors  (a;  —  a),  (a;  — 6),  •••  (a;  —  Q,  real,  unequal 
and  of  the  first  degree. 

Assume  m  =  _A_+B^^...jL.  (1) 

(jiix)     x  —  a     x  —  b     X  —  c      X  —  I 

in  which  A,  B,  O,  •••  L  are  undetermined  coefficients.  Clearing  (1)  of 
fractions, 

f{x)  =  A(x  -b)(x  —  c)  •••  (x-l)-{-  B(x  -  a){x-c)"'(x-l)-] 

+  L(x-a){x-  6) ... (x -  k).  (2) 

Performing  the  indicated  operations  in  (2)  and  equating  the  coeffi- 
cients of  like  powers  of  x  in  the  two  members  by  the  Principle  of 

154 


INTEGRATION   OF  RATIONAL   FRACTIONS.  155 

Undetermined  Coefficients,  will  give  n  equations  from  which  A^  B,  C, 
etc.,  may  be  obtained. 

Or  since  (2)  is  true  for  all  values  of  x,  a  may  be  substituted  for  x, 
which  gives 

(a-b)(a-c)'-'(a-l)  ^  ^ 

By  substituting  b  for  x,  the  value  of  B  is  obtained,  and  so  on; 
finally  when  I  is  substituted  for  x,  it  follows  that 

L  = m (4) 

These  values  of  A,  B,  C,  etc.,  are  substituted  in  (1),  dx  is  intro- 
duced as  a  factor  in  each  terra,  and  each  term  is  then  integrated. 


PROBLEMS. 


1.   Find    rf +  f-^^te. 
J  XT  -\-  XT  —  (yx 


x^-\-x^-6x  =  x{x-\-  S)(x  -  2). 

Assume  f +  f-l    ^A^^G^ 

a^-^x^-Qx      X      x-\-3      x-2  ^^ 

Therefore  x^  +  x-l  =  A{x  +  S)(x  -  2) -]- Bx  (x  -  2) -{-  Cx{x-\-3). 
Substituting  x  =  0,      gives  —1  =  —  6A]  hence  A  =  ^. 
Substituting  x  =  —  3,  gives       5=155;     hence  B  =  ^, 
Substituting  a?  =  2,      gives       5  =  10(7;      hence  C=  J. 

Substituting  these  values  of  A,  B  and  C  in  (1),  introducing  dx,  and 
taking  the  integral  of  each  member, 

dx 


rj^±x-i_.     1  rdx    ,  r  dx      ,r 

=  ^loga;  +  ilog(a;  +  3)  +  ilog(a;-2) 
=  log  [a;^  (a; +  3)^  (a; -2)^]. 
f^^^^  =  log[(x-inx-^2yi 


156  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

J  a'  -  b'x"     2  ah        \a  ~  hxj 

\      r{2x  +  ^)dx^.        (x-l)i   ^ 
J  ^  +  x'-2x  x^{x  +  2Y 

^        r(2  +  ^X-^X^)dx       ,       r    ^/o    .      nI/o  NT 

Case  2.   When  the  denominator  can  be  resolved  into  n  real  and 
equal  factors  of  the  first  degree. 

Let  ^^-Vi:  dx  represent  a  rational  fraction  whose  denominator  can  be 
<^  {^) 

resolved  into  n  factors  each  equal  to  a;  —  a.  In  this  case  the  method 
of  decomposition  of  the  preceding  case  is  not  applicable.     Take,  for 

example,  - — ^. 
{x  —  af 

Forming  the  partial  fractions  as  before  would  give 

2x'-{-x^    A      ^      B      ^      C  .^. 

{x  —  of     X  —  ax  —  ax— a 

But  if  the  fractions  in  the  second  member  are  added, 

27?-\-x^A  +  B+C^ 
{x  —  ay  X  —  a      ^ 

in  which  A  +  B  -\-  C  must  be  regarded  as  a  single  constant,  and  evi- 
dently (2)  cannot  be  an  identical  equation,  as  this  would  give  three 
independent  equations  containing  but  one  quantity,  ^  +  5  +  O,  to  be 
determined. 

The  partial  fractions  are  assumed  as  follows : 

/M^^!_+      -g      +      G      ^..._A_.        (1) 

<^  {x)      {x  -  ay     (x  -  a)"-^     (x  -  ay-^  (a;  -  a)  ^  ^ 

Clearing  (1)  of  fractions, 

f{x)  =  A  +  B{x-a)^C{x-  ay+"-L{x-ay-\  (2) 


INTEGRATION  OF   RATIONAL   FRACTIONS.  157 

The  values  of  A,  B,  C,  etc.,  in  (2),  are  found  by  the  Principle  of 
Undetermined  Coefficients,  then  substituted  in  (1),  after  which  dx  is 
introduced,  and  each  term  is  integrated  separately. 


PROBLEMS. 


1.    Find    C^^^^dx 
J  (x  —  1  r 


{x-iy 

Assume         ^l+i  ^  _A_  +  ^^  +  _CL_ . 

(x-if    {x-if^{x-iy^(x-i) 

Hence  x" -\-l  =  A  + B(x-1) -\- C(x -ly 

=  A+Bx-B^Cx^-2Cx+a 
Therefore  0=1,  5-2C  =  0,  and  A-B^C=^1\ 

whence  C  =  1,  5=2,  and  yl  =  2. 

Therefore    (^^^^^dx=  f-l^^  C^^^+  C^^ 

^      r(^x^-2)dx  ^12x^1^ 

J      {x-\-2Y  {X  +  2Y  ^        ^^    ^    ^ 

When  the  denominator  of  a  rational  fraction  may  be  resolved  into 
both  equal  and  unequal  factors  of  the  first  degree,  the  two  methods 
must  be  combined.  -      . 

4.     r ^ = 5^±12_,  w/x+iV 

J  {x  +  2y{x  +  4y         a^  +  6x  +  S  \x  +  2j 

r_^^zA^±I-dx  =  log  [x(x- 3)^1 

r   dx    ^ ^+JLiog^+^2 


(a^-2y         4(05^-2)     8V2     °  x-^/2 


158  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

Case  3.  When  some  of  the  simple  factors  of  the  denominator  are 
imaginary. 

As  the  denominator  is  real,  the  imaginary  factors  must  occur  in 
pairs,  and  of  the  forms 

••  X  ±a-\-  6V— 1,  and  x  ±a  —  bV—  1, 

whose  product  is  the  real  quadratic  factor 

(x  ±  af  +  b\  (1) 

For  a  single  quadratic  factor  such  as  (1),  the  corresponding  partial 

fraction  will  have  the  form ^t because  each  fraction  of  this 

{x  ±  a)-  +  b^ 
form  increases  by  two  the  degree  of  the  equation  when  it  is  cleared  of 

fractions  and  therefore  increases  by  two  the  number  of  the  equations 

for  determining  A,  B,  C,  .etc. ;  hence  its  numerator  should  add  two  to 

the  number  of  these  undetermined  constants. 

If  the  denominator  contains  n  equal  quadratic  factors,  being  of  the 

form  [(x  ±  ay  +  b'Y, 

the  partial  fractions  may  be  assumed  as  follows; 

f(x)  ^        Ax  -i-  B  Cx-{-  D  Kx-{-L  ^2) 

<t>  (x)  ~  \{x  ±  af  +  b'^Y  ^  lix  ±  af  +  ?>--^]"  '"'  (a.-  ±  of  ^b''         ^  ^ 

The  values  of  A^  B,  C,  etc.,  are  determined  from  (2),  as  in  the  pre- 
ceding cases. 


r 5^ 

J  x^  +  x" 


PROBLEMS. 

+  x'-2 


.  x^  A      ,      B      ,  Cx-\-D 

Hence  ^  =  -i,     B  =  \,     0=0,     i>  =  i. 

/Mx       ^     1  r  dx        1  r  dx       2  r   dx 
x^-^x'-2         qJ  x  +  1      6J  x-1     sJ  0^  +  : 

11      X  —  1  ,  V2 .     -I    X 
=  I  log \-  -^—  tan^ 


y) 


INTEGRATION  OF   RATIONAL   FRACTIONS.  159 

2-     I  7 Tm TT  =  i  2,rc  tan  a;  +  log  ^^ — ^    ^,  • 

3.     f ^^^^ =  logf^!±lf. 

B-  /^';7_^\'^ -  ^  +  log  [(^  +  2)?  (^  -  2)^].  . 

6.     r ^ =ilog^±£±l  +  J_arctan2^±i 

J{x'  +  l){^  +  x  +  \)     ^    ^    x'  +  l     ^V3  V3 


-:  CHAPTER   XVI. 

*-  INTEGRATION  OF  IRRATIONAL  FUNCTIONS. 

Art.  100.     Irrational   Functions. 

Very  few  irrational  functions  are  integrable.  When  an  irrational 
function  cannot  be  directly  integrated  by  one  of  the  elementary  formu- 
las, an  effort  is  made  to  transform  it  into  an  equivalent  rational  func- 
tion of  another  variable  by  making  suitable  substitutions.  When  the 
rationalization  can  be  effected,  the  integral  may  be  found. 

Art.  101.     Irrational  Functions  containing  only  Monomial 

Surds. 

An  irrational  function  containing  only  monomial  surds  may  be 
rationalized  by  substituting  for  the  variable  a  new  variable  with  an 
exponent  equal  to  the  least  common  multiple  of  the  denominators  of 
the  fractional  exponents  in  the  function. 

—dx. 

Assume  x  =  z^',  then  x^  =  ^,  x^  =  z%  and  dx  =  6  z'dz. 

Hence 


rx^  -  2  a 


J      1+Z^  .       J       1   +  ^2 

=  6  Cfz'  -2z'-z'  +  2z'-\-z'-2z-l-}-  ^^^-±l\dz 

=  ^ -  2 z^  -  ^z'  -^  S:^  +  2  z'  -  6  z^  -  ez 

+  6  log(z^  +  1)  -f-  6  arc  tan  z 

=  ^-2x-^x^  -\-Sx^  -\-2x^-6x^  -6x^ 

+  6  log  (x^  +  1)  +  6  arc  tan  xK 
160 


INTEGRATION   OF  IRRATIONAL  FUNCTIONS.  161 

PROBLEMS. 

•^  x^  +  a;3         a;^  ^ 

3.  r^£!^  =  -  isf^  +  ^  + 1|!  +  4«i  +  16.J  +  32  log  (2  -  xJ)l. 

4.  r^illda;  =  -4+^  +  21oga;-241og(a;T2  +  l). 
^  a:3  -(_  x^  ic^      a;T2 

5.  P^'^~^^'da;  =  12(-|a;^  -  f  a;^  +  1^0;^^  _  9^;^) 

+  1908  l^x^^  -  f  a;^  +  3a;^  -  -^x^  +  81a;TiJ  -  243  log  (ojt'^  +  3)]. 


Art.  102.     Functions  containing  only  Binomial  Surds  of  the 

First  Degree. 

m 

A  function  which  involves  no  surd  except  one  of  the  form  (a  +  bxy 
can  be  rationalized  by  assuming  a  -j-bx  =  2;",  as  follows : 


Let /(a;,  va  +  bx)  be  the  function. 


Assume  2  =  V«  +  bx ; 

then  •  z""  =  a  -\-bx, 

n^~Hz  =  b  dx, 


dx  = 
And  x  = 


b 
2"—  a 


Therefore       Cf(x,  ^/a  +  bx)  cZa;  =  ^  Cff^l-^,  z\  z^'-'dz, 
which  is  rational,  and  therefore  can  be  integrated. 

M 


162  DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 


PROBLEMS. 
xdx 


r 

*^  Vl+a; 

Assume  1  +x  =  z^\  then  x  =  z^  —  1,  and  dx  =  2zdz. 

Hence  ("-4^  =  r2{z^-l)zdz  ^  ^  C^,.^,  _  ^,) 


=  12^-22 


^a?  ,  _Vl  +  ic— 1 


2.  r  ^=iog 

*/  JcVl  +  a;  VI  +  a;  + 1 

3.  C{x  •{•■\/x-\-2+-^x-^2)dx 

=  i(a; +  2)2- 2(05  +  2) +|(a;  +  2)^4-f(aJ  + 2)1 

4.  rvT+^a.'  ^  ^^^  ^^  _^  V^TZl)  +  V^"=T. 
*^     V  a;  —  1 

-^  (2r-2/)^ 

g      r     x^dx      ^6a^  +  6a;  +  l 
^  (4a; +  1)^       12(4a;  +  l)^ 

7.     f    ,      ^^  ^8[|(l+Vr3^)^-i(l+Vrr^)^]. 

•>'  Vl+Vl-a; 

Assume  2  =  v  1  +  Vl  —  x. 

s^+^dx 
Art.  103.     Functions  having  the  Form  -,  in  which  n  is 

(a  +  bx'y 

A  Positive  Integer. 

Expressions  of  this  form  may  be  integrated  as  in  Art.  102. 

For  example,  find    I  —         • 

Assume  1  —  ar^  =  2^ ;  then  ar^  =  1  —z^,  and  xdx  =  —  zdz. 


I  INTEGRATION  OF   IRRATIONAL  FUNCTIONS.  163 


Therefore  J^l^  =  - f(lz:^  =  -f(l -z^dz 


=  -(^  -  K)  =  i(l  -  '•O^  -  (1  -  =^)^- 

Art.  104.     Functions  having  the  Form  fix,  J^/^^H-    \^^^ 
In  this  form  the  assumption  may  be  made 
\      that 

then 

I 

'       and 

I       therefore 

The  substitution  of  these  values  will  make  the  function  rational. 


'      \cx  +  d' 

^_ax-\-b 

cx-\-d' 

02"  -  a 

a^-n(ad-bc)z-- 

1 
dz 

(cz^-ay 

PROBLEMS. 

Mx  3ar^  +  2 


if-. 

'^  (l  +  x^)^         3(1 +x2)^ 
•^  V2x2  +  1  3^ 

*   -^  (2  +  3a;2^)^  27(2  +  3ar2)t 

*•    rT-^7^==  =  iiog(V3^^  +  i)  +  }iog(V3:r^~3). 

•^  ar'  +  2V3  —  x^ 


5      r  7^  ~^       da;      _  _  3  3 


3  3 /I -a;' 


(l  +  a;)2  8V'V1  + 


164  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Art.  105.     Functions   containing  only  Trinomial  Surds  of  the 

Form  Va  +  5a;  -f  car^. 

•■ 

Case  1.   When  c  is  positive. 


After  factoring  out  Vc,  the  surd  may  be  written  V-A  +  Bx  +  ar*. 


Assume  VA^i-Bx^i^  =  z  —  x; 

then  A-{-Bx  =  z'^-2zx,  x  =  tjnA^ 

'         B  +  2z 

and  ^^^2(.-+^.  +  ^)c^., 

Therefore     V^  +  ^a;  +  ar^  =  ^  - '^'~  ^  =^' +  ^^  +  A 

jB  +  22  2zH-5 

Thus  the  given  function  may  be  transformed  into  an  equivalent 
rational  function  of  another  variable. 

Case  2.   When  c  is  negative. 

After  taking  out  the  factor  Vc,  the  surd  may  be  written 


WA  -\-Bx-  x\ 
Assume  a  and  /8  to  be  roots  of  the  equation  x^  —  Bx  —  A  =  0\ 


then  Va^  —  Bx  —  A  =  V(a;  —  a){x  —  /8), 


and  V-4  +  ^a;  —  ar^  =  V(a;  —  a){fi  —  x). 


Let  V(a;  —  «)(/?  —  a;)  =  (a;  —  a)  2; ; 

then  (a;  -  a)(P  -  a;)  =  (a;  -  «)V, 

X 


«g^  +  /8 


2^  +  1' 
and  ^^^2.(a-^)d.. 

Therefore       V^  +  5a;  -  ar^  =  (a;  -  a)  z  =  (^~")^ 

r  4-1 

Thus  the  given  surd  is  expressed  in  rational  terms  of  another  vari- 
able. 


INTEGRATION   OF  IRRATIONAL  FUNCTIONS. 


165 


PROBLEMS. 


1.    Find  f ^ 


Bx  +  x" 


Substituting  the  values  of  dx  and  V-4  4-  5aj  +  ar^  as  found  in  Case  1, 
gives 

r dx_ 

^  ■\/A+B, 


_^  r2{z''  +  Bz -\-A) dzx{2z  +  B) 
Bx-{-^    J     iB  +  2zyx(z'  +  Bz  +  A) 


If  5  =  0,  (1)  becomes 


=  logfl  +  »  +  VZ+^T^I 


/: 


dx 


V^  +  ar^ 
If  A  =  ly  (2)  becomes 


=  log[a;  +  V^  +  a^]. 


•^^  Vl  4-ar 

2.   Find  r  ^^ 

^  y/A  +  Bx  —  a? 


(1) 
(2) 

(8) 


Substituting  the  values  of  dx  and  wA  +  5a;  —  a^  as  found  in  Case  2, 
gives 

i  ^A^Bx-^    ^     {f  +  Vf{fi-a)z  Jl+z' 


I 


J  a/s; 


c?a; 


'■/ 


V2  +  3x  +  a:2 
die 


2  arc  cot  f^^ — ^Y 

^dx-G-x"  \^-V 

5.     f— ^?=:=log(i  +  a^  +  V^T^). 


=  —  2  arc  tan  «  =  —  2  arc  tan\/^ — -' 

^  X—  a 

=  log[3  +  2a;  +  2V2T3^+^]. 
3  -  x\^ 


166  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

6.     I ^=1  =  — -  arc  tan  [  —  ^        V 

8.     r^^=  =  '^logf^^±M^\ 

1        /V^+W  +  a\ 
a        \  X  J 


or 


Art.  106.     Binomial  Differentials. 

Binomial  differentials  have  the  form  af  (a  +  hx'^ydXy  in  which  m,  n 
and  p  are  any  numbers,  positive  or  negative,  integral  or  fractional. 
1st.   If  m  and  n  are  fractional,  and  the  differential  has  the  form 

«  I 

ii^(a-{-  hxydx, 

^  =  2**  may  be  substituted,  and  the  expression  becomes 

z'"(a  +  hz^yrtz'^-Mz  =  rtz"-^'''-\a  +  hz^ydz. 

2d.    If  n  is  negative,  and  the  differential  has  the  form 

mf^ia  4-  bx'^ydx, 

X  =  -  may  be  substituted,  and  the  expression  becomes 
z 

x"^  (a  +  bx-'ydx  =  —  2;-"-2  (a  +  bz^dz. 

Hence,  any  binomial  differential  may  be  transformed  into  another, 
having  integers  for  the  exponents  of  the  variable,  and  having  a  positive 
exponent  for  the  variable  within  the  parenthesis. 

In  the  following  articles  every  binomial  differential  is  assumed  to 
have  this  reduced  form. 

Art.  107.     Conditions  of  Integrability  of  Binomial 
Differentials. 


As  the  exponent  of  the  parenthesis  is  any  number,  let  it  be  repre- 

aay  be  written 

p 
x"^  (a -\- bx'^ydx.  (1) 


P 
sented  by  -,  and  the  form  may  be  written 


INTEGRATION  OF  IRRATIONAL  FUNCTIONS.  167 

1st.   Assume  (a  +  6a;")  =  2;* ; 

p 
then  (a  +  6af»)'  =  z^  (2) 


"M 


•-(•!^),  (3) 

1-1 

and  dx  =  -^z^-'l- — ^Y    dz.  (4) 

nh       \     h     J 

Substituting  the  values  from  (2),  (3)  and  (4)  in  (1), 


p 


r^l     ^ 


r"  (a  +  bxydx  =  -^  z^-^'-'  ■- — -]  "      dzj  (6) 

nb  V     ^     / 

which  is  rational  when  — ^t_  is  an  integer  or  0. 


2d.    Assume       aa;~"  +  6  =  2;«;  (1) 

1  1 

then  x  =  or{z!^  —  h)  % 

af'  =  a(:^-b)-\  (2) 

ar  =  a''{z^-b)  ",  (3) 

1  _i_j 

and  da;  =  -^a"(z«-6)  "    H'-^dz.  (4) 

w 

Multiplying  (2)  by  6,  adding  a,  and  taking  the  ?  power, 

(a4-6aJ")'=a'(^'-^)  '2;^.  (5) 

Taking  the  product  of  (3),  (4)  and  (5),  gives 

a;«(a  +  6af')«da;  =  -£«''   «   "(2? -6)  ^"     «    ^z^+'-^da;, 

which  is  rational  when  ^  "^    +  ^  is  an  integer  or  0.     Therefore,  the 

n  q 

binomial  differential  can  be  rationalized : 


168  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

1st.  When  the  exponent  of  the  variable  without  the  parenthesis,  in- 
creased by  one,  is  exactly  divisible  by  the  exponent  of  the  variable  within 
the  parenthesis. 

2d.  When  this  fraction  increased  by  the  exponent  of  the  parenthesis  is 
an  integer. 


PROBLEMS. 


1.   Ym^iC^{2^^x^^dx. 


Here  !!L±1_1  =  ^_1  =  2; 

n  2, 

therefore  the  first  condition  of  integrability  is  satisfied. 
Hence,  let  {2  +  ^^  =  z'^', 

then  (2  +  3a^*  =  2;, 

V  -  2M 


=m' 


zdz 
and  dx = 


27  V7       5        3; 
=  2VR(24-3a;2)|_4(2  +  3a^f 

+  |(2  +  3a^)^]. 

2.  r.-(i+^-^..=(i+4(|^!^. 

%/  oX 

3.  ^x^ia  +  b:^fdx  =  (a  +  b:^^{^  ^"f^'J  ^^» 

4.  f        '^^  f2x  +  ^\l+:^rK 


INTEGRATION  OF  IRRATIONAL  FUNCTIONS.  169 

7.     Ccc^  {a  H-  cc^ydx  =  ^\  (a  +  ic2)'»"  _  s  (^  +  ar^^a  +  f  (a  +  a^^^a^ 

PRACTICAL   PROBLEM. 

A  vessel  in  the  form  of  a  right  circular  cone  is  filled  with  water  and 
placed  with  its  axis  vertical  and  vertex  down.  If  the  height  =  h,  and 
radius  of  the  base  =  r,  how  long  will  it  require  to  empty  itself  through 
kn  orifice  in  the  vertex  of  the  area  a  ? 

Neglecting  the  resistances,  if  the  vessel  is  kept  always  full,  the 
velocity  of  discharge  through  an  orifice  in  the  bottom  is  that  due  to  a 
body  falling  from  a  height  equal  to  the  depth  of  the  water.  If  v 
denotes  the  velocity  and  x  the  depth  of  the  water, 

V  =  V2  gx. 

If  dQ  denotes  the  quantity  discharged  in  the  time  dt  through  an 
orifice  of  the  area  a, 

dQ=adtV2gx. 

But  in  the  time  dt  the  surface  whose  area  is  S  has  descended  the 

distance  dXj  thus 

dQ  =  Sdx. 

Hence  Sdx  =  adt^2  gx^ 

or  dt  = — — 

a-\/2gx 

At  the  distance  x  from  the  vertex. 


'        K^ 


Therefore 


=x 


*   irr'x'dx    ^2  7r?-^Vfe 
dh?^2gx      5aV2g 


CHAPTER   XVII. 

INTEGRATION  BY  PARTS,  AND  BY  SUCCESSIVE  REDUCTION. 

Art.  108.     Integration  by  Parts. 

Integrating  both  members  of  d(uv)  =  udv -^vduy  and  transposing, 
j  udv  =  uv  —  I  V  du.  (1) 

Equation  (1)  is  the  formula  for  integration  by  parts. 
By  this  formula,    I  udv  is  made  to  depend  on    i  v  dii,  and  this  new 
integral  is  frequently  much  simpler  than  the  given  one. 

PROBLEMS. 
1.    Find   I  a^logxdx. 
Let  u  =  logo;,  and  dv  =  x^'dx; 

then  c?w=— ,      and    v=   ^"''' 


X  n-f  1 

Substituting  these  values  in  the  formula  for  integration  by  parts, 

J  n  +  l        n  +  lj 

n  +  lV  n-\-lJ 

2.  I  log  a;da;  =  a;(logaj  —  1). 

3.  Co  sin  0  fie  =  -eGose  +  sin  6. 


170 


INTEGRATION  BY  PARTS.  171 

4.  f!2g22££i^  =  log  ^.  log  (logx)- logo. 

5 .  fxe'^dx  =  €''(-— -\ 
J  \a     ay 

6.  I  arc  sin  a; da;  =  a; arc  sinaj  +  (1  —  a*)*. 

7.  j  a;  cos  a;  da;  =  a;  sin  a; -f  cos  a;. 
xt3in^xdx  =  a;  tan  a;  —  —  -h  log  cos  a;. 

10.     f^e-dx  =  (a^-^-^  +  ^-^-^\^. 
J  \  a       or      a^J  a 

>. 
Art.  109.     Formulas  op  Reduction. 

When  the  integral,  j  x'^{a  +  bx^ydXy 

satisfies  either  of  the  conditions  of  integrability  as  given  in  Art.  107,  it 
may  be  rationalized  as  explained  in  that  article  and  then  integrated. 

But  by  means  of  certain  formulas  of  reduction,  derived  by  the  aid 
of  the  formula  for  integration  by  parts,  the  given  expression  may  be 
made  to  depend  upon  simpler  integrals  of  the  same  form.  This  method 
is  called  integration  by  successive  reduction,  and  the  integrals  given  by 
this  method  are  generally  in  convenient  form  for  integration  between 
limits. 

1.    Formula  A. 
f  Assume        j  x'^{a  -f-  bx'^ydx  =  I  udv  =  uv—  Ivdu.  (1) 

Let  dv  =  x'^'Xa  -f  bxrydx,  then  u  =  af'-*'+\ 

Hence  v  =  (<^  +  ^^Y^     and  du=(m-n-{-l) a;'»-'»c?a?. 

nb(p-\-iy 


172  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Substituting  in  (1),  and  putting  a  +  6x"  =  X, 

fx^X^dx  =  ^'"  '''''^'^'  _  ^  -  ^  +^  fa;— "X^ifia;.  (2) 

•  Now  faj'^-^XP+^da;  =  rx"'-"X^(a  -f-  bx'')dx 

=  a  pc'^-^X^dx  +  b  Cx'^XPdx.  (3) 

Substituting  in  the  right  member  of  (2)  the  value  from  (3), 
C^X'dx  =  ^-'^'X'^l  _  (m  -  n  +  1)  a  r^-„^,^^ 


(p  +  1)         n6(i>  +  l) 
m  —  w  +  1 


JV 


X^dx. 


Transposing  the  last  term  to  the  first  member  and  solving  for  ■  x'^X^dx, 

J  b(np-^m-^l)      b(np  +  m-\-l)J  ^    ' 

By  Formula  (^),  the  given  integral  is  made  to  depend  upon  another 
of  a  similar  form,  having  the  exponent  of  x  without  the  parenthesis 
diminished  by  the  exponent  of  x  within. 

2.   Formula  B. 

=  a  Cx'^X^'^dx  +  b  Cx'^^^'X^-'^dx.  (1) 

Applying  Formula  {A)  to  the  last  term  of  (1),  by  substituting  m  +  n 
for  m,  and  p  —  \  for  />, 

6  fx-x-d^  =  _^!!!^  _  «("»  +  !)  f»"X'-'*., 

•/  np  +  m+1      np  +  m-\-\J 

which  substituted  in  (1),  by  uniting  similar  terms,  gives 

Cx^X'dx  =     ^"'''^''     + ^^ Cx^X^-Hx.  (E)      ; 

J  np  -\-  771  -{•  1      np  -{-  7n  -\-  IJ  - 


INTEGRATION  BY   SUCCESSIVE   REDUCTION.  173 

By  Formula  (B)j  the  given  integral  is  made  to  depend  upon  another 
of  a  similar  form,  having  the  exponent  of  the  parenthesis  diminished 
by  unity. 

Formulas  (A)  and  (B)  fail  when  np  -h  m  +  1  =  0,  but  in  this  case 

^?^^t — \-pz=Oj  hence  the  method  of  integration  of  Art.  107  is  appli- 

n 
cable. 

3.  Formula  O. 

Solving  Formula  (A)  for  j  x'^'^'X^dx,  gives 

C^-X'd^  =    f-"^"'    -  Hnp  +  m  +  l)  r^^,^^ 
J  a{m  —  n-\-l)       a{m  —  n-^T)J 

Substituting  —  m  for  m  —  n, 

C^-«X'dx  =   ^"7'^'^;'   +  Km-n-np-l)  T^-™.,^,^.        (C) 
J  —  a(m  — 1)  —  a(m  — 1)       J 

By  Formula  (C),  the  given  integral  is  made  to  depend  upon  another 
of  a  similar  form,  having  the  exponent  of  x  without  the  parenthesis 
increased  by  the  exponent  of  x  within. 

Formula  (O)  fails  when  m  —  1  =  0 ;  in  this  case 

m  =  1,  and  —  m  +  1  =  0 ; 

hence  the  method  of  integration  of  Art.  107  is  applicable. 

4.  Formula  D. 

Solving  Formula  (J5)  for  j  xT^X^-Hx,  gives 

J  anp  anp         J 

Substituting  —  p  for  p  —  1, 

•/  an(p— 1)  a?i(2>  — 1)       */ 

By  Formula  (Z>),  the  given  integral  is  made  to  depend  upon  another 
of  a  similar  form,  having  the  exponent  of  the  parenthesis  increased  by 
unity. 

Formula  {D)  fails  when  p  —  1  =  0,  but  in  this  case  the  integral 
reduces  to  a  fundamental  form. 


174  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

PROBLEMS. 

1.  Find    f-^^. 

Here  m  =  3,  n  =  2,  p  =  —  ^,  a=  a^,  and  6  =  —  1. 
Substituting  these  values  in  (A), 

3  «j 

2.  Find    f-i^— . 

Here  m  =  0,  w  =  2,  —  p  =  —  3,  a  =  1,  and  6  =  1. 

Substituting  these  values  in  (D), 

Applying  (Z))  to  the  last  term  of  (1),  making  mJtO,  n=2,  — p  =  —  2, 
a  =  1,  and  6  =  1, 

J(l  +  x^-'dx  =  ^(^  +  '^"'  +  ij(l  +  x^r'dx 
=  ;r7r^ — s:  +  i  arc  tan  x. 

Therefore  /(TT^a  =  4irT^^ 
8.   Find    r       ^^       . 

f ^ =  Cx'^a'  -  x^-kx 


PROBLEMS.  175 

Here  —  m  =  —  3,  n  =  2,  p  =  -  J,  a  =  (j?,  and  &  =  —  1. 
Substituting  these  values  in  (0), 

By  Art.  105,  Ex.  8, 


.  a; 
_  arc  sin  — 
2  a 


arc  sm  — 


Therefore  r_^^^==- vV^+   1    log^^^Vo^. 

6.  ("(1  -  iB')^dic  =  Ja;(l  -  ar^^  +  f  a;(l  -  ic^)^  +  f  arc  sina?. 

7.  J'VCa^  +  a^(^^  =  I  V^^M^  + 1  log  {X  +  V^M=^. 

-'vr^^     1^5^5.3^5.3; 

9.     far^Cl  -  a;2)^c?a;  =  ^  a:  (2  a;^  -  1)(1  -  ar^^  +  1  arc  sin  a?. 
10.     fa^^ (1  _^  ^f^a. ^  (^^-2)  (1  +  a^)t. 


12  ■    ^-     /I     " 


13.     f      '^'i^       =.-(^g'  +  g^  +  ^V2ax-r'  +  4a^arcversg- 
-'V2ax-x^         V3         6         27  a 

Kbmabk.   Reduce    f-^^—^  to  the  form    ^ x^  (^  a  -  xf^dx. 
J  ^2  ax -01?  -^ 


176  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

^  ^2ax-x'  2  a 

*-  C    x*dx  x(S  —  xF) 

15.  I 7  =  — ^ T  —  I- arc  sm x. 

J  (l-x')^     2(1 -x")^ 

9 

16.  r_^^  =  _  2  (3a;«  +  4ar»  +  8)Vir:^. 


CHAPTER   XVIII. 

INTEGRATION  OF  TRANSCENDENTAL  FUNCTIONS.      INTEGRATION 

BY  SERIES. 

Art.  110.     Introduction. 

The  method  of  integration  by  parts  gives  important  reduction 
formulas  for  transcendental  functions.  Only  a  comparatively  small 
number  of  logarithmic  and  exponential  functions  can  be  integrated  by 
general  methods.  It  is  frequently  necessary  to  resort  to  methods  of 
approximation.  Some  of  the  principal  integrable  forms  will  be  given 
in  this  chapter. 

Art.  111.     Integration  of  the  Form    i  f(x)(logxydx. 

It  is  assumed  in  this  form  that  f(x)  is  an  algebraic  function  and  n 
is  a  positive  integer. 

Let  f(x)dx  =  dVf  and  (log  a?)**  =  u ; 

then  Cf(x)dx  =  v,  and  n  (log  xy-^  —  =  du. 

J  X 

Substituting  these  values  in  \udv  =  uv—  \v du, 

Cf{x)  (log  xydx  =  (log  xyCfix)  dx  -  C[n  (log  xy-^  —  Cf(x)  dx\ ; 

or,  by  making  j  f(x)  dx  =  X, 

Cf{x)(iogxydx=X(iogxy  -  n  C—(\ogxy-Hx.  (1) 

Hence,  whenever  it  is  possible  to  integrate  the  factor  f(x)dx,  the 
given  integral  will  depend  upon  another  of  a  similar  form,  in  which 
the  exponent  of  the  logarithm 'is  diminished  by  unity.     By  repeated 
N  177 


ITS  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

applications  of  this  formula  the  given  integral  will  depend  finally  on 
the  algebraic  form    |  <j>(x)dx. 

PROBLEMS. 
1.    Find   Cxilogxfdx. 

Let  xdx  =  dv,  and  (log  xy  =  Wy 

oiy  dx 

then  —  =  '^)  and  2  log  x  —  =  du. 

2  x 


dx 

X 


Hence  i  a;  (log  xydx  =  —  (log  xy—  I  x^  log  x  ■ 

Similarly,         |  x  log  xdx=z—  (log x)—  i • 

J  2  J  2   X 

/Qiyt                  y3.               a^     ' 
X  (log  xydx  =  —  (log  xy  —  —  log  X-] 

2 .     fx* (log ic)2 ^a;  =  ^ (log^ x-^logx-}-  j\). 
rlogxdx     xlogx  ,  1      /-,        N 

5.     raf'(logx)^dai=  ^[(logxy 2     loga;  +  -^1. 

c/  71 H- 1 1_  71  +  1  (n  +  1)^J 

Art.  112.     Integratiol'?  of  the  Form    |  x"*a*^c?a;. 

In  this  form  it  is  assumed  that  m  is  a  positive  integer. 
Let  a'^dx  =  dv,  and  a;"*  =  w ; 

then  -— =  V,  and  mx'^'^dx  =  dw. 

w  log  a 

Therefore        Cx'^a'^dx  =  _^!^ !??_  far^-ia'^'da;. 

*/  nloga     nloga*/ 


INTEGRATION   OF   TRANSCENDENTAL   FUNCTIONS.       179 

By  successive  applications  of  this  formula,  the  exponent  of  x  can 
be  finally  reduced  to  zero,  and  the  given  integral  made  to  depend  on 

the  known  form,    I  a'^'dx. 

PROBLEMS. 

1.  Find    Cx^e'^'dx. 

Let  e'^'^dx  =  dv,  and  a;^  =  w ; 

then  (  rcV'^dic  =  -  e'^^'x^ (  xe'"dx. 

J  a  aJ 

Similarly,  j  xe'^'^dx  =  -  e^^o; (  e'^^'dx. 

J  a  aJ 

Therefore  Cx'e'^dx  =  —  fx''  -  ^'  +  ^\ 

J  a  \  a       (Tj 

2.  ^xa'dx    =J^(x  — 


log  a  \        log  a^ 

3.  Cx^e'dx  =e'{x'^  -2x+2). 

4.  fa^e-da;  ==  ~(:x?  _  5  a^  -f- 1  a;  -  -,\ 
J  a\        a         or        (rj 

5.  r^      =_e-(ic2-|.2a;  +  2). 


Art.  113.     Integration  of  the  For:.i    i  sin**  ^  cos"  ^  d^. 


'■/■ 


1st.    When  either  m  or  ti,  or  both,  are  odd  positive  integers. 
In  this  case  the  integration  can   be   effected  as  in  the   following 
example :  ^, 

fsin^  (9  cos-*  (9  dd  =  C(l  -  cos^  0)  cos*  0  sin  0  dO 


=  —  i  (cos^  6  —  cos*'  0)  d  cos  0 


cos^     cos^ 


180  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

2d.    When  m  -\-  n  is  an  even  negative  integer. 
In  this  case  the  integration  can  be  effected  as  in  the  following 
example : 

fsin^  e  cos-^  edO=  ftan^  0  cos"*  0  dO 
=  Aan^^sec^^d^ 
=  Aan^  ^  (1  +  tan^  0)  •  d  tan  ^ 

=  Atan^  ^  .  d  tan  ^  4-  tan^  ^  .  d  tan  6) 

^tan^     tan^ 
3  5' 

3d.   When  the  form  is  not  immediately  integrable,  neither  of  the 
aforesaid  conditions  being  fulfilled. 

In  this  case  the  integral  can  be  obtained  by  successive  reductions. 


Let  sin  ^  =  ic ;  then  sin"*  6  =  x"",  cos"  ^  =  (1  —  af)\ 

and  de  =  (1  -  x'y^dx. 

Hence  fsin"*  0  cos"  0  dO  =  fx"'  (1  -  af)~^dx.  (1) 

\ 

Thus  the  given  trigonometric  form  may  be  transformed  into  a 
binomial  differential  which  may  be  integrated  by  means  of  the  formulas 
of  reduction.  | 

For  example,  to  find    |  sin*  0  cos*  0  dO. 
Let  sin  ^  =  a;  J  then  sin*  0  =  x\  cos*  ^  =  (1  —  x^^^  and 
de  =  (l-  x^y^dx.  ^ 

Hence  fsin*^  cos*  0  dO  =  Cx\l  -  af)^dx. 

Applying  Formula  (A)  twice, 
fx\l  -  x-)idx=-  ^H^^  _  I .  Jx(l  -^i  +  i-  if(l  -  ^^dx.  (2) 


INTEGRATION  OF   TRANSCENDENTAL   FUNCTIONS.       181 

Applying  Formula  (B)  twice  to  the  last  term  of  (2), 

Hence 

+  I  •  i  •  i  •  i^(l  -  «^^^  +  f  •  i  •  f  .  i  arc  sin  a;. 
Therefore 

fsin^  0  cos*  6de  =  -  ^—^  (sin^  ^  +  i  sin  ^)  +  ^i^  sin  6  (cos^  ^  +  f  cos  ^) 

PROBLEMS. 

cos^^     cos^^  .  cos^^'^' 


..     fsin^^  cos^  ^d^  =  -  [  ^^^^  -  -^^  + 


6  4      ■      10 


2.  f-^  =  tan  ^  +  I  tan^^  +  i  tan^^. 
J  cos^  0 

3.  f!H^!^  =  sec  ^  +  2  cos  ^  -  4  cos^^. 

4      r       de        ^         1         _  4cos^  _  8  cos  e 
J  sin*  0  cos^  ^  ~  cos  ^  sin^  ^     3  sin^  ^     3  sin  6 

5 .     fsin^  0  cos^  edO=\  sin*  ^  -  |  sin«  d. 

g      rsin'^^^^^tan^(9     tan^^ 
J  cos«^  5  3* 

7 .  fsin* Ode  =  -\cosO (sin^ ^  +  f  sin ^)  + 1 ^. 

8.  fcos* edO  =  ^  sine (cos^ ^  +  |  cos ^)-  +  | ^. 

«       r      "zi    •   4^^/i      sin  ^  cos  ^ /sin* ^      sin^^      1\  ,    6^ 

11.     rsec^ede  =  52£l*22i  +  |log(secfl  +  tan«). 
J  sm^  ^  2  sm^  ^  2 


182  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Art.  114.     Iijtegration  of  the  Forms    j  x""  cos  (ax)  dx  and 
j  X"  sin  (ax)  dx. 

The  formula  for  integration  by  parts  is  used,  assuming  that  u  =  ic". 
Eviciently,  each  application  of  the  formula  will  diminish  the  exponent 
n  by  one ;  therefore,  when  n  is  a  positive  integer,  the  given  form  may 
be  made  to  depend  finally  on  j  sin  (ax)  dx  or  j  cos  (ax)  dx,  each  being 
a  simple  known  form. 

For  example,  to  find  |  x^  sin  ic  dx. 

Assume  u  =  x^,  and  dv  =  sin  x  dx ; 

then  du  =  2x dx,  and  v  =  —  cos x. 

Hence        j  a^  sin  a;dx  =  —  a^  cos  «  +  2  j  .1; cos  a;da;. 

Similarly,  I  x cos xdx  =  x sin x—  i  sin x dx 

=  X  sin  X  +  cos  X. 
Therefore  1 0? suixdx  =  —  a? cos x  -\-  2 xmix  -\-  2 cos x. 

Art.  115.     Integration  of  the  Forms  I  6*"=  sin" a; da;  and  |  e**cos"a;da;. 
Assume  u  =  sin**  x,  and  dv  =  e'^dx-, 

then  du  =  n  sin""^  x  cos  xdx,  and  v  =  — 

a 

Hence      |  e**  sin**  xdx  =  -  e*^  sin**  a;  —  -  j  e"''  sin**~^  x  cos  a;  da;.  (1) 

J  a  aJ 

Again,  assume  u  =  sin""^  x  cos  a;,  and  dv  =  e*""  da; ; 

then  du  =  (n  —  1)  sin**"^  a;  cos'^  x  dx  —  sin**  x  dx 

=  (n  —  1)  sin**"*^  xdx  —  n  sin**  a;  da;, 

and  v  =  — 

a 


INTEGRATION  OF  TRANSCENDENTAL  FUNCTIONS.       183 

Hence 

/ga*  sinn-i  ^  QQQ  xdx=  -  e**  sin"~^  x  cos  x ^^—  I  e"*  sin"~^  x  dx 
a  a    J 

+  -  fe**  sin"  a;  da;. 
aJ 

Substituting  in  (1),  and  solving  for  I  6*"=  sin"  x  da;,  gives 

/._   .   „     ,        e"'  sin""^  X  (a  sin  x  —  n  cos  x) 
e"''  sm"  xdx  = ^-— — ^ 
n^  +  a" 

+  ^'^f  ""i^  fe"*  sin"-  2  a;  dx,  (2) 

n^  +  a^  J 

Each  application  of  this  formula  diminishes  the  exponent  of  sin  x 
by  2.  By  repeated  applications  n  can  be  reduced  to  0  or  1,  and  the 
given  integral  will  finally  depend  on 

I  e^'dx  =  — ,  or  j  e'"'  sin  xdx. 
The  value  of  j  e***  sin  a;  da;  is  obtained  directly  from  (2)  by  making 

In  like  manner  J  e**  cos"  05  da;  can  be  obtained. 


n  =  l. 


PROBLEMS. 


1.  j  ar' cos  a;  da;  =  ar*  sin  a; +  2  a;  cos  a;— 2  sin  a;. 

2.  I  a^  sin  a;  da;  ==  —  x^  cos  a;  +  3a;2sina;  +  6a;  cos  x  —  6  sin  x. 

e"*  sin  xdx  =  — (a  sin  a;  —  cos  x). 

4.  j  e*  sin^  a; da;  =  —  (sin^  a;  +  3  cos^  a;  +  3  sin  a;  —  6  cos  a;). 

r  are       2     7        6"'=  COS  a;  (a  cos  a;  +  2  sin  x) 

5.  I  6°="  cos^  a; da;  = ^^- -^ ^ 

J  4  +  a2 


2        e^ 


184  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Art.  116.     Integration  of  the  Forms 
j  f{x)  arc  sin  x  dx,      I  f(x)  arc  cos  ic  dx,      j  f(x)  arc  tan  x  dx,  etc. 

In  these  forms,  f(x)  is  an  algebraic  function. 

Any  one  of  these  forms  may  be  integrated  by  using  the  formula  for 
integration  by  parts,  assuming  dv  =  f(x)  dx.     For  example,  to  find 


/' 


ar*  arc  sinxdx. 


Assume  dv  =  Mx,  and     u  =  arc  sin  x ; 

then  1?  =  — >       and  du  = 


3  Vn^ 

y? ■  ,  r    Mx 


Hence  |  oc^  arc  sin  xdx  =  —  arc  sin  x  —  ^  i  - 


Vl^-^ 
Substituting  a  =  1,  in  Ex.  1,  Art.  109, 

^  Vl  —  ar 
Therefore    j  y?  arc  ^wvxdx  =  —  arc  sin  a?  +  J (1  —  ar^^(aj^  -}-  2). 

PROBLEMS. 

1.  I  arc  sin  xdx  =  x  arc  sin  aj  -f-(l  —  a^)^. 

2.  I ^ — =  X  arc  tan  x  —  ^  (arc  tan  «)^—  ^  log  (1  -f  a^. 

-      /*a^  arc  sina;daj  ^ /-i  ,  n\    r^ 2  •        ,  a;^  ,    , 

3.  I —  =  — -^(3^  +  2)  Vl —  ar' arc  sin  a? -h  —  +  |a;. 

*/      Vl  —  ^  9 

4.  la;  arc  cos  xdx  =  ^ar^  arc  cos  a;  —  J  a;(l  —  qi?Y  +  \  arc  sin  a?. 




f— ^?— =  f- 

»/  a  -f-  6  cos  B  ~  J     I 


a[cos^g)+sin^(|)]f.[ 


6cos^ 
dO 

COS'*!  -  1—  sm-*  - 


INTEGRATION   OF   TRANSCENDENTAL   FUNCTIONS.       185 

dO 


-f 


(a  +  b)cos'(^)  +  {a-b)sm'(^ 


=/; 


sec-Y^W^ 


(a  +  &)  +  (a-6)taii2(| 


-/; 


^6 
d tan  f  - 


(a  +  6)  +  (a-5)tan2^|^ 

^        arc  tanff ^^^y tan  (^\],  when  a  >  6.     (1) 


li  a<b,  then 


d  tan' 


w  V6  +a+V6  —  atanj  -  j 

=  VP^^  ^"^  -== ^= 7^'  (2) 

by  Ex.  4,  Art.  99. 


V6  +  a— V6  —  atan(-j 


In  like  manner   ( ^.       may  be  obtained. 

J  a  -\-b  sm  a; 


1.    Find 


fi 


PROBLEMS. 

dO 


2  4-  cos  ^ 
Substituting  a  =  2  and  6  =  1  in  (1)  gives 

(Vitan|). 


r     rZ^             2 

1 = arc 

J2  +  cos^      V3 

tan 

r     d$      _ 

J  3  +  5  cos  (9 

tan|  +  2 
tan|-2 

■/^ 

dO 
-  4  cos  2  ^ 

=  1  arc  tan  (3  tan 

X). 

186  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

Art.  118.     Integration  by  Series. 

When  a  given  function  cannot  be  integrated  by  any  of  the  preced- 
ing methods,  or  when  the  integral  obtained  is  too  complicated  in  form 
for,  convenient  use,  recourse  is  had  to  the  method  of  approximation 
called  integration  by  series.  By  this  method  the  function  is  developed 
into  a  series  whose  terms  are  integrated  separately.  If  the  resultant 
series  is  convergent,  an  approximate  value  of  the  integral  is  found  by 
summing  a  finite  number  of  terms. 

PROBLEMS. 
1,   Find  Cx'(l-x')^dx. 

Expanding  (1  —  a^)^  by  the  Binomial  Theorem, 
Therefore  Co? (1  -  a?)^  da  =  Cx'{l  -  |ar'  -  |a;<  -^i)fi—)dx 


__a^      ^       x' 
3      10     56 

144 

'  dx               a?     a?     a;*     ar^ 
1  +  x            2"^3      4"^6     ' 

3.     Cx^{l-x^^dx=,^x^-\x^--^x^-^x^ 

.      r     dx  a^    ^     3ar*  3.5a;^ 

4-      I  —  =  X—  --— --t- 


VIT^         2.3    2.4.5    2.4.6-7 


-e'x" 


dx. 


VI -ar^ 

Therefore 

dx 


Vl-^ 


J  vr=^      J\  2.4       2.4.6       J 

=  arc  sin  aj  -f  ^  e^  (^  x  Vl  —  a:^—  ^  arc  sin x) 
-2^e''r(ia^  +  i-}a^)Vr^-|^arcsina;T... 

6.     ra;~^(a;-  1)' fZa;  =  f  a;^  -  4  a;*  +  3^^  a;"^  4.  ^aj'V" ..-. 


^ 


{1 


CHAPTER   XIX. 

INTEGRATION  AS  A  SUMMATION.     AREAS  AND  LENGTHS  OF 
PLANE  CURVES. 


Art.  119.     Integration  as  a  Summation. 

The  integration  of  a  function  may  be  regarded  as  the  summation  of 
a  certain  infinite  series  of  infinitely  small  terms.  The  problem  of 
finding  the  areas  of  plane  curves  furnishes  a  good  illustration.* 

For  example,  to  find  the  area  AP^P^N,  included  between  the  curve 
BS,  the  X-axis,  and  the  ordinates  AP^  and  NP^. 


y 

Pn 

-.Pn                  .             . 

„     P. 

^ 

— o 

R 

> 

^3 

^ 

/ 

0 

/ 

\    \ 

3    C 

:  c 

) 

- 

t 

4                         X 

Fig.  32. 

Let  y  =^f(x)  be  the  equation  of  the  curve.  And  let  OA  =  a,  0N=  6, 
and  divide  AN  into  n  equal  parts  each  denoted  by  Aa;,  and  erect  ordi- 
nates at  the  points  of  division. 

Then,  area  of  rectangle  Pj-B     =/(a)Aa;, 

area  of  rectangle  P2C     =f{Gb  +  Aic)  Aa;, 

*  Newton's  Lemma  XL,  Principia^  Lib.  I.,  §  1. 
187 


188  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

area  of  rectangle  P^D     =/(a -f2  Aic)Aa;, 

aifd  area  of  rectangle  P^_.iN=f(b  —  Aa;)Aa;. 

[jDheref ore,  the  sum  of  the  n  rectangles  is 
f(a)Ax  -\-f{a  +  Aa;)  Aa;  +/(a  +  2  Aa^)  Aa;  +  ...  +/(5  -  Aa;)Aa;,     (1) 

which  may  be  represented  by  ^    /(a;)Aa;,  in  which /(a;)  Aa;  represents 

each  term  of  the  series,  x  taking  in  succession  the  different  values 
between  a  and  h. 

Now  as  Aa;  approaches  zero,  n  increases  indefinitely,  and  the  limit 
of  the  sum  of  the  rectangles  is  the  required  area  AP^P^N. 

When  Aa;  becomes  dx^  the  symbol  ^     is  replaced  by   I  ,  and  the 

expression  for  the  area,  which  is  the  sum  of  an  infinite  number  of 
infinitely  small  rectangles,  becomes 

C f{x)dxz=f{a)dx-^f{a-\-dx)dx+f{a-{-2dx)dX'-.-\-fQ)  —  dx)dx.  (2) 

Assume  I  f(x)  dx=  <f>  (x) ; 

then  /  (x)  dx  =  d<f>  {x)  =  <f>(x  ■{-  dx)  —  <f>(x).  (3) 

Substituting  successively  for  x  in  (3)  the  values, 
a,  a  -j-  dx,  a  -\-2  dx,  -•'  b  —  dx, 

gives  "  ■"""-=/"(a)t?a;     - ~(^(a-\-    dx)  —  <}i(a),  .---—. 

f(a-\-  dx)  dx    =  <j>(a-{-2  dx)  —  <^  (a  +  dx), 
f(a  +  2  dx)dx  =  cf> (a  -{-  3  dx)  —  KJi  (a  -\-  2  dx), 

f(h-dx)dx     =<f>(b)-<li(b-dx). 
Adding  these  equations, 

f(a)dx-\-f(a-hdx)dx-\-"'f(b  —  dx)dx=:<f>(b)  —  <l>(a)y 

or  r  f(x) dx  =  <f>{b)—<f> (a). 

Therefore  the  area  is  found  by  integrating  f(x)dx,  substituting  b 
and  a  successively  in  the  integral,  and  subtracting  the  latter  result 
from  the  former. 


INTEGRATION   AS  A  SUMMATION. 


189 


—  arc  sm  - 
2  rjo 


PROBLEMS. 
1.   Find  the  area  of  the  circle  x^  -\-y^  =  r^. 

Area  of  a  quadrant  =  j   f(x)dx=  I   ydx=.l    (r^  —  a^^dx. 
By  Ex.  4,  Art.  109, 

~  4   ' 

therefoi^e  the  area  of  the  whole  circle  is  ttt^. 

In  order  to  obtain  the  area  of  the 
semi-segment  OABC,  Fig.  33,  the 
superior  limit  of  integration  will  be 
OA  =  X,  and  the  inferior  limit  will 
be  zero. 

Therefore- 


area  OABC  =  C  (r"  -  x^^dx 


■^ — - — ^  +  —  arc  sm  -• 
2  2  r 


Fig.  33. 


Evidently 


xjr'-x'y 


area  of  triangle  OAB, 


and 


—arc  sin  -  =  area  of  sector  OBC. 
2  r 


2.    Find  the  area  between  the  curve  y^ 
ordinate  through  the  focus. 


and  X  =  1. 


4:X,  the  axis  of  X,  and  the 
Ans.  A  =  ^. 

3.  Find  the  area  of  the  ellipse  a^  +  6V  =  a^b^.  Ans.  -n-ab. 

4.  Find  the  area  of  the  hyperbola  xy  =  1  between  the  limits  a;  =  a 

Ans.  Log  a. 

In  this  example  it  will  be  seen  that  the  area  of  the  hyperbola  is  the  Naperian 
logarithm  of  the  superior  limit.  For  this  reason  Naperian  logarithms  are  also 
called  hyperbolic  logarithms. 


190 


DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 


5.   Find  the  area  of  the  cycloid  x  =  rare  vers  -  —  \/2  ry  —  y\ 

dx=    y^y   . 


Here 

Therefore 


■W^ry-y^ 


=^j: 


fdy 


Sttt^. 


W2ry-y^ 

6.    To  find  the  area  of  y  (l-\-x^)  =^1  —  o?,  between  the  curve  and 
the  axes,  in  the  first  quadrant. 

The  limits  will  be  found  to  be  a;  =  1  and  a;  =  0. 

1-0^ 


Therefore 


+  0^ 


dx 


Jo  [_  or^  +  lar'  +  lj 

r      ^  HI 

=    --  +  ilog(a;2  +  l)4-arctana; 

=  .631972. 
7.   Find  the  area  included  between  y^  =  2px  and  oi?  =  2py, 

yI 


Fig.  34. 


The  two  parabolas  intersect  at  (0,  0)  and  (2p,  2p) ;  hence  the  limits 
of  integration  are  2p  and  0. 

Area  OBPA=  f  V2pxdx. 


Area  OCPA 


=1 


2p 


dx. 


INTEGRATION  AS  A   SUMMATION.  191 

Therefore  area  OBPC  =  TY V2^  -^\dx  =  4^'. 

8.  Find  tlie  area  included  between  y^  =  2x  and  y^  =  4:X  —  x^. 

Ans.  0.475. 

9.  Find  the  entire  area  within  the  hypocycloid  x^  -{-  y^  =  a^. 

Ans.  ^^ 
8 

10.  What  is  the  area  of  a  theoretical  indicator  diagram  when  the 
steam  is  cut  off  at  half-stroke,  if  the  law  of  expansion  is  jj^y  =  1  ? 

A71S.  1  +  log  2. 

Art.  120.     Areas  of  Plane  Curves  in  Polar  Coordinates. 

Referring  to  Fig.  35,  it  is  required  to  find  the  area  POP^,  included 
between  any  plane  curve  AB  and  two  vectors  OP  and  OP^. 

B> 


Fiw.  o5. 


Let  the  vectorial  angles  POX  and  PnOX  be  denoted  respectively 
by  y8  and  a. 

If  the  coordinates  of  any  point  P  be  (r,  0),  then  the  coordinates  of 
Pj  will  be  (r  +  Ar,  0+^0). 


The  area  of  sector  POS  =  i  r  •  rAO  =  i  r^AO. 


Then  the  sum  of  all  the  sectors  POS,  PiOSi,  etc.,  may  be  repre- 
sented by  ^     ^7-^  Ad\  and  as  A^  approaches  zero,  the  limit  of  the  sum 

of  the  sectors  is  the  required  area  POP^,  which  will  be  given  by  the 

expression  ^^ 

A  =  ^  \    rhW. 


192  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

PROBLEMS. 

1 .    Find  the  area  of  the  logarithmic  spiral  r  =  a^,  between  the 

limits  ?'2  and  r^. 

"     Here  dr  =  a»  log  a  dO,  and  dO  =     ^^'^    ♦ 

rloga 

' •  Hence  A=^  C-" rhW  =  -^—  C\ dr  =  f-^T' 

Vri  2  log  « Jr,  |_4  log  ajrj 


4  log  a 

2.  Find  the  area  described  by  one  revolution  of  the  radius  vector 
of  the  spiral  of  Archimedes  ?'  =  aO.  a         j  _  i  T^"  22^^  _  4:7rW 

—  Vo    ^  ""       "■  ~3 

3.  Find  the  area  of  the  lemniscate  1^  =  o?  cos  2  ^.  ^?is.  a^ 

4.  Find  the  area  of  a  loop  of  the  curve  r  =  a  cos  2  ^.         ^?is.  i  tto?. 

5.  Find  the  entire  area  of  the  cardioid  r  =  a(l  —  cos 6).  q     2 

2 

6.  Find  the  area  of  a  loop  of  the  curve  r^  cos  0  =  a?  sin  3  ^. 

^ris.  ^-^'log2. 
4        2^ 

Art.  121.     Rectification  of  Plane  Curves  referred  to 
Rectangular  Axes. 

By  Art.  72, '  '   ds  =  [l  +/'^YTda;, 

in  which  s  represents  the  length  of  the  arc. 

Therefore  « =f  [l  +  (|)]*  cT. ;  (1) 

the  limits  of  integration  being  the  limiting  values  of  a?. 

The  process  of  finding  the  length  of  an  arc  of  a  curve  is  called  the 
rectification  of  the  curve. 

If  y  be  considered  the  independent  variable,  the  formula  is 

•-f['-(S)T*  <»> 

in  which  d  and  c  are  the  limiting  values  of  y.  -.  .  ■ 


AREAS  AND  LENGTHS  OF  PLANE  CURVES. 


193 


If  the  arc  PR,  in  Fig.  36,  is  to  be  rectified,  the  value  of  the  first 
derivative  is  found  from  the  equation  of  the  curve,  and  substituted  in 


Fia.  36. 


Formula  (1)  or  (2).  If  Formula  (1)  is  used,  the  limits  b  and  a  are  OB 
and  OA  respectively ;  if  Formula  (2)  is  used,  the  limits  d  and  c  are  OD 
and  OG  respectively.  ^ 

PROBLEMS. 

1.   Rectify  the  parabola  '(f  =  2px. 


Here 

dy     p. 
dx     2/' 

ice 

dx=y^y. 

P 

Therefore 

s=Cfl 

= mV^  +1  log  (2,  +  ^^Tf)  +  a         (1) 

Here  the  value  of  the  constant  C  may  be  determined  by  the  first 
method  of  Art.  34.  If  the  arc  is  estimated  from  the  origin,  then  /S  =  0 
when  2/  =  0,  and  these  values  substituted  in  (1)  give 


O  =  |logi>+Ci 


hence 

Therefore 


O: 


P 


logp. 


yVfTf  _^p  1    (y^-^p'  +  y\ 

2p  2       \  p  / 


194  DIFFERENTIAL   AND   INTEGRAL  CALCULUS. 

which  is  the  length  from  the  vertex  to  the  point  which  has  the  ordi- 
nate y. 

Or  if  the  limits  of  integration  are  known,  for  instance,  if  the  length 
from  the  vertex  to  an  extremity  of  the  latus-rectum  is  required,  then 
the  limits  are  p  and  0,  and 


s  = 


Vp'  +  y\E  log  (y  +  Vp'  +  /)T  =  ip  V2  +g  log (1  +  V2). 


2 


2p  2 

2.  *  Rectify  the  semi-cubical  parabola  y^  =  aa^. 

Ans.  -I-A +|aa;Y--^. 
27  a\       ^     J       21a 

3.  Rectify  the  curve  whose  equation  is  y"^  =  —y  and  determine  the 

length  of  the  curve  from  the  origin  to  the  point  whose  ordinate  is  10. 

Ans.  19.0248. 

4.  Rectify  the  circle  ar^  -j-  2/^  =  i-^. 

Here  g  =  4  TYl  4-^^(^a;=:4r  T      ^^       =27r7\ 

Jo  \^       2/7  ^^  vV  -  x" 

But  as  the  result  is  in  circular  measure,  the  circle  is  a  non-rectifiable 
curve. 

An  approximate  result  may  be  obtained  by  a  series. 

A     r      d^  A    [^  .      ^      ,     1  •  3  x^     ,      "!'■ 

'='U  ^j^^  =  ^'\j^2:^?^2riTw?^'''i 
L  ^2.3^2.4.5^2.4.6.7^  J 


Therefore 


L       2.3     2.4.5     2.4.6-7         J 


From  this  equation  the  approximate-  value  of  ir  can  be  determined 
with  any  required  degree  of  accuracy  by  taking  a  sufficient  number  of 
terms. 

5.   Rectify  the  ellipse  2/^  =(1  —  e^){a?  —  x^. 


Here  ^  =  _  (1  _  ,^)  ?  =  _  ^^^Izi^^. 

dx  y         Va^-ar^ 

*  The  semi-cubical  parabola  was  the  first  curve  whose  rectification  wa>  effected 
algebraically.     (Neil,  in  1657,  Phil.  Trans.,  1673.) 


AREAS  AND   LENGTHS  OF   PLANE   CURVES.  195 


Hence  s  =  4  CJ^Lf^dx  =  4.r  —^ — (a^  -  e'x^^ 

Jo    M  a^-  x^  Jo   Va^  -  x^ 

J«   V^^^^V        2a      2.4a3      '" J 

6.  Rectify  the  hypocycloid  x^  -}-y^  =  a^. 

Ans.  aS  =  I  a^x^ ;  the  entire  curve  =  6  a. 

7.  Rectify  the  cycloid  x  =  r  arc  vers  -  —  V2  ry  —  y^. 

Here  ^=         ^        • 

^2/      V2ry-y^ 

Therefore  s  =  2  f  Y^r^^ V  di/  =  8  r. 

Jo    \2r-yJ 

Art.  122.     Rectification  of  Curves  in  Polar  Coordinates. 

By  Art.  73  (2), 

-[-HI)']'*- 
Ti,,...„    -r['*+(i)T'"- 

PROBLEMS. 
1.   To  find  the  length  of  the  cardioid  r  =  a  (1  +  cos  0). 

Here  ^  =  -asin^. 

dO 

Therefore  ^  =  ^  f"  f ^'  ^^  +  ^°^  ^)'  +  ^' ^^^'  ^]^  ^^ 

=  2a  C\2-j-2cose)^de  =  4:a  CcosidS 
c/o  c/0  2 

=    8  a  sin  -      =  8  a. 


196  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

2.    Eectify  the  spiral  of  Archimedes  r  =  aO. 


2a        ^2     ^\  a  J 


3.    Rectify  the  logarithmic   spiral   logr  =  ^  between  the  limits  Vi 
and  r-o.  Ans.  (1  -f-  m^)^(ri  —  Vo). 


4.    Rectify  the  curve  r  =  a  sin' 


0 


Ans. 


Art.  123.     The  Common  Catenary. 

The  common  catenary  is  the  curve  assumed  by  a  flexible  cord  of 
uniform  thickness  and  density,  fastened  at  two  points,  hanging  freely 
and  acted  upon  only  by  the  force  of  gravity. 

As  the  cord  is  regarded  as  perfectly  flexible,  the  only  force  acting 
at  any  point  of  the  cord  is  a  pull  in  the  direction  of  the  cord  at  that 
point,  which  is  called  tension  and  is  a  function  of  the  coordinates  of 
that  point. 


Y 

\        ^ 

/ 

v 

/' 

0 
N 

/ 

X" 

0' 

Fia.  37. 


In  Fig.  37,  let  0,  the  lowest  point  of  the  curve,  be  the  origin,  and 
let  the  horizontal  line  through  0  be  the  X-axis,  and  the  vertical  OY 
be  the  F-axis.  Let  (x,  y)  be  any  point  P  on  the  curve,  s  the  length  of 
OP,  and  c  the  length  of  the  cord  whose  weight  is  equal  to  the  tension 
at  O. 

If  the  weight  of  the  unit  of  length  be  taken  as  the  unit  of  weight, 
the  length  s  will  represent  the  weight  of  the  arc  OPf  and  the  length  c 
will  represent  the  tension  at  0. 


AREAS   AND   LENGTHS  OF   PLANE  CURVES.  197 

Then  the  arc  OP  may  be  regarded  as  a  rigid  body  in  equilibrium 
under  three  forces :  the  tension  at  P  in  the  direction  of  the  tangent, 
the  horizontal  tension  c  at  the  origin,  and  the  weight  s  acting  verti- 
cally downward. 

Draw  PN  tangent  to  the  curve,  and  PS  parallel  to  OX  at  P.  Then 
by  the  triangle  of  forces,  the  sides  of  the  triangle  PSN  will  represent 
the  three  forces  acting  on  the  arc  OP. 

Therefore  |g  ^  weight  of  OP     ; 

SP       tension  at  O       c 

hence  -^  =  -.  (1) 

dx     c 

Differentiating  (1),  substituting  the  value  of  ds,  and  reducing, 


\/'+v* 


dx 


dy\-       c 


(2) 


Integrating  (2)  and  noticing  that  when  a;  =  0,  -^  =  0, 

dx 


'-[lW'H2)]=v 


or  .    »,      .  I 

dx       ^         \dx 


whence  -^=i(e<=  — e  *). 

dx     ^^  '         • 

Integrating  (3),  and  noticing  that  a;  =  0  when  y  =  0, 


(3) 


y  =  'L{^  +  e'')-c,  (4) 

which  is  the  required  equation. 

Removing  the  origin  to  the  point  0',  which  is  at  a  distance  c  below 
0,  the  equation  becomes 

3,  =  |(e^  +  e')-  («) 


198  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

In  order  to  rectify  the  catenary,  (5)  is  differentiated,  giving 
|  =  i(J_e-j),  asm(3), 
from  which  is  obtained 

Therefore  s  =  ^  C\e^  +  e~'']dx  =  |(e'  -  e~}. 

PROBLEM. 

What  is  the  curve  in  which  the  cables  of  a  suspension  bridge  hang  ? 

Ans.  A  parabola. 


CHAPTER   XX. 


SURFACES  AND  VOLUMES  OF  SOLIDS. 


Art.  124.     Surfaces  and  Volumes  of  Solids  of  Revolution. 


1st.  Surfaces.  In  Fig.  38,  let  the  plane  curve  MN  revolve  about 
the  X-axis.  Let  M  be  a  fixed  point  and  P  any  other  point  of  the  curve 
whose  coordinates  are  {x,  y). 


/ 

< 

5      0 

A 

^ 

-R 

^N 

C 

A       B 
Fig.  38. 


Assume  MP  —  s  and  PQ  =  As,  then  the  coordinates  of  Q  are 
(x  -\-  Ax,  y  +  Ay).  Let  S  represent  the  area  of  the  surface  generated 
by  the  revolution  of  MP,  and  AS  the  surface  generated  by  PQ.  Draw 
PR  and  QT,  each  equal  in  length  to  As,  and  parallel  to  OX.  In  the 
revolution  PR  generates  the  convex  surface  of  a  cylinder  whose  area 
is  2  Try  As,  and  QT  generates  the  convex  surface  of  a  cylinder  whose 
area  is  2  tt  (2/  -f  Ay)  As.  Obviously  the  area  of  the  surface  generated  by 
PQ  lies  between  the  areas  of  the  cylindrical  surfaces. 


Hence 


2  7r2/  As  <  AaS  <  2  77  (2/  H-  Ay)  As. 


Therefore,  as  As  approaches  zero. 


dS  =  2iry  ds, 
199 


200  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

and  S=C2  7ryds  (1) 

dx.  (2) 


=  2./,[ 


1  +  '^'^ 


dx 


^  In  like  manner  for  the  surface  generated  by  revolving  the  curve 
about  the  F-axis, 

S  =  2  wCx  ds  =  2  ttCx  ["l  +  (^)'f  ^y-  (3) 

The  surface  of  a  zone,  included  between  two  planes  perpendicular 
to  the  X-axis  and  corresponding  to  the  abscissas  6  and  a,  is 

=  2  7rjjds.  .  (4) 


S 


2d.  Volumes.  Let  V  denote  the  volume  generated  by  the  surface  in- 
cluded by  the  curve  MP,  the  ordinates  ML  and  PA,  and  the  X-axis. 

Let  A  V  represent  the  volume  generated  by  APQB. 

The  volume  of  the  cylinder  generated  by  APGB  is  iry^AXf  and  the 
volume  of  the  cylinder  generated  by  ASQB  is  ir(2/  -f-  AyY^x. 

Obviously,  iry^Ax  <  A  F<  tt  (?/  +  /^yyAx. 

Therefore,  as  Aa;  approaches  zero, 

dV=  Try^dx, 

and  V=  TT I  y^dx.  (5) 

In  like  manner,  for  the  volume  generated  by  revolving  the  curve 
about  the  F-axis, 

V=7r  Ca^dy.  '  (6) 

PROBLEMS.  I 

1.   Find  the  surface  of  the  sphere  generated  by  revolving  the  circle 

x^  -{-y^  =z  r^  about  a  diameter.  J 

Here  y=(7^-a^)^  and  —  =  --• 

(XX         y 


SURFACES  AND  VOLUMES  OF   SOLIDS.  201 

-Therefore   S=2Tr  ry(l-{-^^  dx  =  2'jrr  C  dx^  4.-^7^. 

2.  Find  the  volume  generated  by  revolving  the  parabola  y^  —  2px 
about  the  X-axis. 

V=  TT  i    2pxdx  =pTrx^  =  1  tt/  •  X. 

3.  Find  the  volume  of  the  cone  generated  by  revolving  y  =  x  tan  a, 
when  a  is  the  semi-vertical  angle  of  the  cone. 

Ans.   V=\  volume  of  circumscribing  cylinder. 

4.  Find  the  volume  of  the  sphere  generated  by  revolving  oi^-{-y-=r^ 
about  the  X-axis,  and  also  the  volume  of  a  spherical  segment  between 
two  parallel  planes  at  distances  b  and  a  from  the  centre. 

Ans.  J TTT^  and  tt  [r2(6  -  a)  -  ^  (6»  -  c^)]. 

5.  Find  the  surface  and  volume  of  the  prolate  spheroid  generated 
by  revolving  y^  =  (1  —  e^{a?  —  x')  about  the  X-axis. 

Ans.  S  =  27r¥  -] arc  sin  e  and  V  —  — - — 

e  3 

6.  Find  the  surface  and  volume  of  the  right  circular  cone,  gener- 
ated by  revolving  the  line  joining  the  origin  with  the  point  (a,  b)  about 

the  X-axis.  /  ^  ,  ,  n       ,    ^r     'rrab^ 

Ans.  S  =  7rb  Va*  -h  b^  and  V=  ——. 

S 

7.  Find  the  surface  generated  by  the  cycloid 


y  =  r  arc  vers  -  -f-  V 2  rx  —  oc^, 
r 

when  it  revolves  about  its  axis.  *  Ans.  8  irr^iv  —  f ). 

8.  Find  the  volume  generated  by  the  cycloid 

a;  =  r  arc  vers  -  —  V2  ry  —  y\ 

T 

when  it  revolves  about  its  base.  Ans.  5  ttV. 

9.  Find  the  surface  and  volume  of  the  annular  torus,  generated  by 
revolving  the  circle  a?  -\- {y  —  5"y  =  4",  about  the  X-axis. 

Ans.  S  =  394.79  sq.  in.,  and  V=  394.79  cu.  in. 


202 


DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


Art.  125.     Surfaces  by  Double  Integration. 

In  Fig.  39,  let  {x,  y,  z)  and  (x  -{- dx,  y  -\-  dy^  z  +  dz)  be  the  coordinates 
of  two  consecutive  points  P  and  E  on  the  given  surface  whose  equation 
is  known.  Through  P  and  E  pass  planes  parallel  to  the  planes  XZ 
and  *YZ.  These  planes  will  intercept  an  element  PE  of  the  curved 
surface,  which  is  projected  on  the  XY^plane  in  BS  =  dxdy. 


Fig.  39. 


Let  S  represent  the  required  area,  and  dS  the  area  of  the  element 
PE. 

The  area  of  PS  is  evidently  equal  to  the  area  of  PEj  multiplied  by 
the  cosine  of  the  angle  which  PE  makes  with  XY. 

Representing  this  angle  by  y. 


hence 


area  PE  •  cos  y  =  dxdy^ 
area  PE  =  dxdy  •  sec y. 


By  the  aid  of  analytical  geometry  of  three  dimensions,* 

in  which  —  and  —  are  partial  derivatives  from  the  equation  of  the 

dx  dy  ^  ^ 

given  surface. 

*  See  Appendix ;  Note  A. 


SURFACES  AND  VOLUMES  OF  SOLIDS.  203 

Therefore   area  FE  =  dS  =  fl  -\-f^\-^f^YYdxdyy 

The  effect  of  the  2/-integration,  x  remaining  constant,  will  be  to  give 
the  sum  of  all  the  elements  similar  to  PE  from  W  to  M:  hence  the 


limits  of  the  ^-integration  will  he  y  =  CM=^^ OM'^—xr  and  y  =  0. 

The  effect  of  the  subsequent  a;-integration  will  be  to  give  the  sum 
of  all  the  strips  similar  to  WMN  forming  the  given  surface ;  hence  the 
limits  of  the  second  integration  are  x  =  OA  and  a;  =  0. 

Art.  126.     Volumes  by  Triple  Integration. 

The  given  volume  is  supposed  to  be  divided  into  elementary 
rectangular  parallelopipeds  by  planes  parallel  to  the  three  coordinate 
planes ;  such  an  element  of  volume  is  represented  by  kl  in  Fig.  39. 

The  volume  of  such  an  elementary  parallelopiped  is  dx  dy  dz ;  hence 
the  whole  volume  is 

F=  C  C  Cdxdydz.  (1) 

The  limits  of  integration  are  obtained  from  the  equation  of  the 
bounding  surface,  being  so  chosen  as  to  embrace  the  entire  volume. 

If  the  volume  included  between  the  three  coordinate  planes  and  the 
curved  surface  is  required,  the  limits  are  found  as  follows : 

The  effect  of  the  ^-integration  is  to  sum  all  the  elemental  parallelo- 
pipeds in  the  prism  PS-,  hence  the  limits  of  the  first  integration  are 
PR  =  z  =f(x,  y)  and  2  =  0.  The  effect  of  the  ^/-integration  is  to  sum 
all  the  elemental  prisms  in  the  slice  WN\  hence  the  limits  for  the 
second  integration  are  CM=y=f(x)  and  y  =  0.  The  effect  of  the 
a;-integration  is  to  sum  all  the  elemental  slices  composing  the  whole 
volume ;  hence  the  limits  of  the  third  integration  are  x  —  OA  and  x  =  0. 

Ou       ?y       z 
For  example,  to  find  the  voli^ne  of  the  ellipsoid  —  -f  ^  +  —  =  1,  cut 

or     Ir     (f 

off  by  the  coordinate  planes.  -  


Here  the  limits  of  the  2^-integration  are  c a/1 i~   2  ^^^  ^?  *^® 


204  DIFFERENTIAL  AND   INTEGRAL  CALCULUS. 


limits  of  the  y-integration  are  6  -vl ^  and  0,  and  the  limits  of  the 

ic-integration  are  a  and  0. 

"^2         ~2  j  ^ 

1 ;;  —  r:;,  and  yi  =  b\l in  the  formula, 

a^     b^  ^        a? 

gives  for  the  whole  ellipsoid 

9 

/•a     r»y\    f*tx 

F=  8  j      1      \     dx  dy  dz. 
Integrating  on  the  hypothesis  that  z  is  the  only  variable, 

Integrating  again,  now  on  the  hypothesis  that  y  is  the  only  variable, 

Integrating  finally  with  respect  to  x,  gives 

V='^^r{a^-x^dx=^^abc, 

PROBLEMS. 
1.   Find  the  surface  of  the  sphere  qi?  -\- y^  -\- z^  =.  a*. 

Here  |2=  _2,  and  |^  =  -?!. 

dx         z  oy         z 

dxdy 


Therefore    5=  JJ(l  +  J  + J)^.x.,=//_^ 


a^-/ 


Integrating  with  respect  to  i/,  between  y  =  Va^  —  3?  and  y  =  0, 


SURFACES  AND   VOLUMES   OF   SOLIDS.  205 

Integrating  with  respect  to  «,  between  x  =  a  and  a;  =  0, 

Jo    2  2 

which  is  the  area  of  one-eighth  of  the  surface  of  the  sphere. 


2.  A  sphere  x^ -\- y^ -j- z^  =  a~  is  cut  by  a  right  circular  cylinder 
y-  =  ax  —  x^.  Find  the  area  of  the  surface  of  the  sphere  intercepted 
by  the  cylinder.  Ans.  2a^(7r  —  2). 

3.  Find  the  surface  intercepted  by  two  right  circular  cylinders 
a^-\-z^=  a^  and  x'  -{-  y^  =  a^.  Ans.  8  al 

4.  Find  the  volume  of  a  right  elliptic  cylinder  whose  axis  coin- 
cides with  the  X-axis  and  whose  altitude  =  2  a,  the  equation  of  the 
base  being  c^  +  b^z'^  =  b^c-.  Ans.  2  irabc. 

5.  Find  the  volume  of  the  solid  contained  between  the  paraboloid 
of  revolution  a^-\-y^  =  2zj  the  cylinder  ar^  +  y^  =  4  a;,  and  the  plane  z  =  0. 


xi+yi 

Ans.  2pP'''"'f   '    dx  clydz^  37. 699- 

«/0    »/0  t/0 


6.  Find  the  volume  of  the  solid  cut  from  the  cylinder  a^-\-y^  =  a^ 
by  the  planes  z  =  0  and  z  =  x  tan  a.  Ans.  f  a^  tan  a. 

7.  Find  the  entire  volume  bounded  by  the  surface  x"^ -\- y^ -\- z^ 
=:-s/2E.  Ans.  44.88. 


CHAPTER   XXI. 

CENTRE   OF  MASS.     MOMENT  OF  INERTIA.     PROPERTIES  OF  GULDIN. 
Art.  127.     Definitions. 

The  definitions  of  this  article  are  taken  from  Mechanics  and  are 
here  assumed  without  investigation. 

The  moment  of  any  force  with  respect  to  an  axis  perpendicular  to 
its  line  of  direction  is  the  product  of  the  magnitude  of  the  force  by 
the  perpendicular  distance  from  its  line  of  direction  to  the  axis.  The 
moment  of  a  force  with  respect  to  a  plane  parallel  to  its  line  of  direc- 
tion is  the  product  of  the  force  by  the  perpendicular  distance  from  its 
line  of  direction  to  the  plane. 

The  force  exerted  by  gravity  on  any  body  is  proportional  to  the 
mass  of  the  body,  and  hence  the  mass  of  the  body  may  be  taken  as  the 
measure  of  the  force  exerted  on  it  by  gravity. 

The  centre  of  mass  of  a  body  is  that  point  so  situated  that  the 
force  of  gravity  produces  no  tendency  in  the  body  to  rotate  about  any 
line  passing  through  the  point ;  hence  it  may  be  regarded  as  the  point 
at  which  the  whole  weight  of  the  body  acts.  The  centre  of  mass  is 
sometimes  called  centre  of  gravity  and  centre  of  inertia. 

The  moment  of  inertia  of  a  body  with  reference  to  a  straight  line, 
or  plane,  is  the  sum  of  the  products  obtained  by  multiplying  the  mass 
of  each  element  of  the  body  by  the  square  of  its  distance  from  the 
line  or  plane. 

Points,  lines  and  surfaces,  as  here  considered,  are  supposed  to  be 
material  bodies.  Lines,  surfaces  and  solids  are  regarded  as  being  com- 
posed of  an  infinitely  large  number  of  indefinitely  small  particles.  The 
weight  of  a  body  is  the  resultant  of  the  weights  of  all  of  its  elemental 
particles  acting  in  vertical  lines,  and  the  resultant  of  this  system  of 
parallel  forces  passes  through  the  centre  of  mass. 

206 


CENTRE   OF   MASS. 


207 


Art.  128.     General  Formulas  for  Centre  of  Mass. 

Assume  a  system  of  rectangular  coordinate  axes,  retaining  a  fixed 
position  with  reference  to  the  body,  the  plane  XY  being  horizontal. 
Let  a  small  particle  of  mass  at  any  point  (x,  y,  z)  be  represented  by 
Am.  Then  the  force  exerted  by  gravity  on  Am  is  measured  by  Am  in 
a  direction  parallel  to  the  Z-axis. 


Fm.  40. 

^f  the  mass  of  Am  were  concentrated  at  the  point  (x,  y,  z),  the 
moment  of  the  force  exerted  on  Am  with  respect  to  the  plane  YZ 
would  be  icAm ;  and  the  sum  of  the  moments  of  all  the  elements  of  the 
body  with  reference  to  this  plane  would  be  2a;Am. 

The  resultant  force  of  gravity  is  ^SAwi,  and  if  the  coordinates  of  the 
centre  of  mass  be  represented  by  (x,  y,  z),  as  the  centre  of  mass  is  the 
point  through  which  the  resultant  passes,  S^Am  will  be  the  moment  of 
the  resultant  with  reference  to  the  plane  YZ.  But  by  the  principle  of 
moments,  the  moment  of  the  resultant  of  any  number  of  forces  is  equal 
to  the  algebraic  sum  of  the  moments  of  the  forces. 

Hence,  x^\m  =  2a;Am. 


If  now  Am  diminishes  indefinitely, 

X  I  dm  =  I  xdm. 

I  xdm 

Therefore  x='^ . 

I  dm 


(1) 


208  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

I  ydm 

Similarly,  ^^^r '  ^^^ 

j  dm 

/zdm 
.-— (3) 

I  dm 

The  mass  of  any  homogeneous  body  is  the  product  of  its  volume 
by  its  density.  If  k  represents  the  constant  density  and  dv  the  ele- 
ment of  volume,  then  kdv  =  dm,  and  (1),  (2)  and  (3)  become 

I  xdv 
^=^.  (4) 

iydv 

I  zdv 

and  ^='V-'  (^) 

fdv 

If  the  body  is  a  material  line  in  the  form  of  the  arc  of  any  curve, 
and  if  ds  is  the  length  of  an  element  of  the  curve.  Formulas  (4),  (5) 
and  (6)  become 

I  xds 
^=^'  (7) 

fas 

Cyda 
y=J—~,  (8) 


zds 

fas 

If  the  curve  is  a  plane  curve,  it  may  be  taken  in  the  plane  XT,  in 
which  case  z  will  be  zero. 


CENTRE  OF  MASS. 


209 


PROBLEMS. 


1.   Find  the  centre  of  mass  of  an  arc  of  a  circle,  taking  the  diam- 
eter bisecting  the  arc  as  the  X-axis  and  the  left  vertex  as  the  origin. 
In  Fig.  41,  let  AOB  be  the  arc. 


Fia.  41. 


The  equation  of  the  circle  is  ^  =  2  aa;  —  sc*; 


hence 


dy 


_  (a  —  x)dx 
V2  ax  —  sc^ 


ds  =  Vdoc^  H-  dy^  = 


adx 


f^^^     a  r' 
Therefore  x=^ =  -  I        

Cds       ^^'^  V2aa;-ar^     « 


V2  ax  —  x^ 
xdx       =«(_V2^^3^  +  s)=a-52^. 


2.   Find  the  centre  of  mass  of  an  arc  of  the  hypocycloid 

x^  -\-y^  =  aJ, 
between  two  successive  cusps  : 

Here 


dy  =  -(^'^dxi 


hence 


ds  =  -Vda^  H-  dy^  =  y^^x'^  -{-y  ^dx 


=  V  tt*  -  x^ 


V--*  +  -T^ 


■dx  =  (-]^dx. 


a^  —  x^ 


210 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


Therefore 


Similarly, 


OM==x  = 


SHtf" 


Jo    \xj 


dx 


=  i' 


Fig.  42. 


fa. 


3.    Find  the  centre  of  mass  of  the  arc  of  a  semi-cycloid. 

Ans.  x=  (Tr  —  ^)a,  y  = 

Art.  129.     Centre  of  Mass  of  Plane  Surfaces.* 
If  rectangular  coordinates  are  used, 
dv  =  dA  —  dxdy, 
and  Formulas  (4)  and  (5)  of  Art.  128  become 

I    I  xdxdy 

\    {  dxdy 

\    \  ydxdy 

y=itLtL 

I   I  dxdy 
The  centre  of  mass  of  a  plane  area  is  sometimes  called  the  centroid  of  the 


and 


(1) 


(2) 


CENTRE   OF  MASS. 


211 


PROBLEMS. 

1.    Eind  the  centre  of  mass  of  the  area  included  between  the  parab- 
ola if-  =  2px  and  the  double  ordinate  whose  abscissa  is  a. 


I     xdxdy       I    xydx 

.  __      0   J-y =z^ 

I    dxdy         I    ydx 

0    J~y  Jo 

r*a        3 

I       ^2j)X'-i 
_./0 


■dx 


x^dx 


I  a. 


2.    Find  the  centre  of  mass  of   the   semicircle  y?-\-y^  =  r^  on  the 
right  of  the  F-axis. 


I     I    xdxdy       I    xydx 

(     (  "dxdy         i  'ydx 
Ja  J  y  Jo 


T^dx 


I     V?"  —  x^dx 


4r 


3.    Find  the  centre  of  mass  of  an  elliptic  quadrant  whose  equation 
b 


IS 


a 


.       -      4a    -      46 
Ans.  X  =    — ,  y  =  - — 

3  TT  O  TT 


4.  In  Fig.  43,  ABD  is  a  segment  of  a  parabola  cut  off  by  an  ordi- 
nate, and  BE  is  parallel  to  Ax. 

1st.  Determine  the  distance  of  the  centre 
of  mass  of  ABD  from  Ax.  .        3y 

8 

2d.  Determine  the  centre  of  mass  of 
ABE.  ,        (Zx    3y 

5.  Find  the  centre  of  mass  of  the  cycloid. 


Ans.  X  =  Trr,  2/  =  f  r. 


212  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

6.   Find  the  centre  of  mass  between  ?/"*  =  a;"  and  ?/"  =  x"". 

Ans.  x  =  y  = (^+^y 

(m  +  2n)(2m  +  n) 


,Art.  130.    'Centre  of  Mass  of  Surfaces  of  Revolution. 

If  a  curve  in  a  plane  with  the  X-axis  be  revolved  about  this  axis,  then 

dv  =  2'iryds\ 
hence,  by  Art.  128  (4), 

I  2irxyds       I  xyds 
i  *2-jryds        iy  ds 

PROBLEMS. 

1.   Find  the  centre  of  mass  of  the  convex  surface  of  a  right  cone, 
generated  by  the  line  y  =  ax.      . 


Here  ds  =  ^dx^  +  dy^  =  Vci^  -r  ^dx; 

I  ax^  Vci^  -i-ldx 

hence  '^  =  ~^ ZZHZT —  =  i^- 

I  ax  Va^  -{-  Idx 

2.   Find  the  centre  of  mass  of  the  surface  generated  by  the  revolu- 

'fj         

tion  of  a  semi-cycloid  x  =  a  vers~^  -  —  V2  ay  —  y^  about  its  base. 


Here  ds  = 


__  V2ady 
V2  a  —  y 

p     xydy 

hence  x=  ^^^      =||a. 

3.   Find  the  centre  of  mass  of  the  convex  surface  of  a  hemisphere 
whose  radius  is  equal  to  10.  Ans.  x  =  5. 


CENTRE  OF  MASS.  213 

Art.  131.     Centre  of  Mass  of  Solids  of  Eevolution. 

If  a  solid  be  generated  by  the  revolution  of  a  plane  curve  about 

the  X-axis,  then 

dv  =  2irydydx\ 
hence,  by  Art.  128  (4), 

I   j  xydxdy 
jjydxdy 

PROBLEMS. 

1.   Find  the  centre  of  mass  of  a  right  circular  cone,  whose  convex 
surface  is  generated  by  revolving  y  =  ax  about  the  X-axis. 


•*'  /*x    /•ax 

J.  X  """ 


dx 


—dx 


2.   Find  the  centre  of  mass  of  a  paraboloid  generated  by  ^  =  4:  ax. 

n'*' xydxdy 
- 


x  =  ' 


ff'^'ydxdy 

J  I     2aMx 
=  |aj. 

J-»x  <* 

2axdx 

3.   Find  the  centre  of  mass  of  a  hemispheroid  generated  by 

52 
y^  =  —  (2ax  —  x^.  Ans.  -fa. 

CL 


214 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Art.  132.     Moments  of  Inertia  of  Surfaces. 

In  Fig.  44,  the  curves  AB  and  CD  and  the  ordinates  LN  and  PM 
intercept  a  plane  surface  PLSR,  whose  moment  of  inertia  is  required. 

The  surface  is  supposed  to  be  divided  into  rectangular  elements  by 
lines  parallel  to  the  coordinate  axes. 


f 

D 

J 

)  - 

r 

L 

k-^^ 
c^^^^ 

" 

~ 

-.^^^^^ 

n 

^^^■^B 

f 

J 

J 

R         W  V 

^ 

^^--0 

Fig.  44. 


Let  (a;,  y)  be  the  coordinates  of  any  point  as  /,  then  (x  -{-  dx,  y  +  dy) 
will  be  the  coordinates  of  g^  and  dxdy  will  be  the  area  of  the  element^. 

The  moment  of  fg  about  X=  y^dxdy. 

Let  the  equations  of  AB  and  CD  be  ?/  =  f{x)  and  y=<f>(x)  respec- 
tively ;  and  let  0N=  b  and  OM—a. 

If  X  be  regarded  as  constant,  while  y  varies  from  <^  (x)  to  /(a;),  the 
integration  will  give  the  moment  of  the  vertical  strip  WQTV. 

Then  in  the  second  integration,  x  varying  from  a  to  h,  the  sum  of 
the  moments  of  all  the  stripe  composing  the  area  PRSL  will  be  given. 

Representing  the  moment  of  inertia  by  M.  I., 


fdxdy 


PROBLEMS. 
1.  Find  the  moment  of  inertia  of  a  circle  about  its  diameter. 


M.I.  =  (     I    '^  *^ yHxdy 

*/-rt/-v'r2-i2 


PROPERTIES  OF  GULDIN.  215 

2.  Find  the  moment  of  inertia  of  a  rectangle  about  an  axis  through 
its  centre  parallel  to  one  of  its  sides. 

Let  2  h  and  2  d  denote  the  width  and  length  respectively,  the  axis 
being  parallel  to  h ;  then 


M.  I.  =  C  Cy^dx  dy  =  ^  hd\ 


3.  Find  the  moment  of  inertia  of  an  isosceles  triangle  about  an 
axis  which  passes  through  its  vertex  and  bisects  its  base. 

Let  a  =  the  altitude  and  2  6  =  base,  and  take  the  origin  at  the 
vertex  and  the  axis  of  moments  as  the  X-axis ;  then 


-X 


Art.  133.     Guldin's  Theorems.* 

I.   Let  a  plane  curve  in  the  same  plane  with  the  X-axis  revolve 
about  the  X-axis. 

The  ordinate  of  the  centre  of  mass  is 


^Jyds 


-,  by  Art.  128  (8). 


Therefore 


2'jry  '  s  —  2ir  {yds.  (1) 


But  by  Art.  124  (1),  the  second  member  of  (1)  is  the  area  of  the 
surface  generated  by  the  revolution  of  the  curve  whose  length  is  s 
about  the  X-axis,  and  the  first  member  is  the  circumference  described 
by  the  centre  of  mass,  multiplied  by  the  length  of  the  curve  s. 

Hence,  if  a  plane  curve  revolve  about  an  axis  in  its  own  plane 
external  to  itself,  the  area  of  the  surface  generated  is  equal  to  the 
length  of  the  revolving  curve,  multiplied  by  the  circumference  de- 
scribed by  its  centre  of  mass. 

*  Sometimes  called  Theorems  of  Pappus,  as  they  were  first  stated  by  Pappus. 


216  DIFFERENTIAL    AND    INTEGRAL   CALCULUS. 

II.  A  plane  area  revolves  about  the  X-axis.  The  ordinate  of  the 
centre  of  mass  of  the  plane  surface  is 

•-1  11  ijdxdy 

y=-Y-r '     by  Art.  129  (2). 

•;  ijdxdy 

Therefore        ^"ry  \   I  dxdy  =  27r  I  I  ydxdy  =:7r  i  yHx.  (2) 

But  by  Art.  124  (5),  the  last  member  of  (2)  is  the  volume  gener- 
ated by  the  revolution  of  the  area ;  and  in  the  first  member,  I  i  dxdy 
is  the  revolving  area.  Hence,  if  a  plane  area  revolve  about  an  axis 
external  to  itself,  the  volume  generated  is  equal  to  the  area  of  the 
revolving  figure,  multiplied  by  the  circumference  described  by  its 
centre  of  mass. 

If  the  curve  or  area  revolve  through  any  angle  0  instead  of  making 
an  entire  revolution,  6  must  be  substituted  for  27r  in  equations  (1) 
And  (2). 

PROBLEMS. 

1.  Find  the  surface  and  volume  of  the  ring  generated  by  revolving 
a  circle  whose  radius  =  r,  about  an  external  axis  distant  &  from  the 
centre  of  the  circle.  Aii&.  S  =  ^  n^ah^    V=  2  7rV6. 

2-  Find  the  volume  generated  by  an  ellipse  revolved  about  an  axis 
distant  10  from  the  centre ;  the  semi-axes  being  10  and  5. 

Ans.  9869.6+. 

3.  Find  the  surface  and  volume  generated  by  revolving  a  cycloid 
about  its  base.  Ans.  S  =  -^Tra',  V=  5v^a\ 


CHAPTER   XXII. 

DIFFERENTIAL  EQUATIONS. 

Art.  134.     Definitions. 

A  differential  equation  between  two  variables  x  and  y  is  an  equation 
containing  one  or  both  of  the  variables  x  and  y  and  one  or  more  deriva- 
tives, such  as  -^,  — ^,  —^,  etc. 
dx  dxr  dor 

The  order  of  a  differential  equation  is  that  of  the  highest  derivative 
which  it  contains. 

The  degree  of  a  differential  equation  is  that  of  the  highest  power 
to  which  the  highest  derivative  which  it  contains  is  raised. 

The  solution  of  a  differential  equation  consists  in  finding  a  relation 
between  x  and  y  and  constants,  from  which  the  given  equation  may  be 
derived  by  differentiation;  this  relation  is  called  the  primitive.  The 
solution  requires  one  or  more  integrations,  and  each  integration  intro- 
duces an  arbitrary  constant ;  hence  the  solution  of  a  differential  equa- 
tion of  the  nth  order  will  give  an  equation  containing  n  arbitrary 
constants. 

The  same  primitive  may  have  several  differential  equations  of  the 
same  order. 

For  example,  given  the  equation 

ay -\- bx -\- c  =  0.  (1) 

By  differentiating,  a^i-b  =  0.  (2) 

dx 


Eliminating  a  between  (1)  and  (2), 

bx^  +  cf- 
dx       dx 

Eliminating  b  between  (1)  and  (2) 


2,a;^-hc^-62/  =  0.  (3) 

dx       dx 


ay-^c-ax^  =  0.  (4) 

dx 

217 


218  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

In  this  example,  equation  (1)  is  called  the  complete  primitive,  and 
equations  (2),  (3)  and  (4)  are  differential  equations  showing  the  same 
relation  between  the  variables. 


A?iT.  135.     Differential  Equations  of  the  First  Order  and 

Degree. 

The  general  form  of  the  differential  equation  of  the  first  order  and 

degree  is 

Mdx  +  Ndy  =  0,  (1) 

in  which  Jtf  and  ^  are  functions  of  x  and  y.     This  equation  may  be 
put  in  the  form 

ax 

The  most  obvious  method  of  solving  a  differential  equation  of  the 
first  order  and  degree  is  by  means  of  the  separation  of  the  variables, 
whenever  practicable.  The  variables  are  separated  when  the  coefficient 
of  dx  contains  the  variable  x  only,  and  the  coefficient  of  dy  contains  the 
variable  y  only ;  that  is,  when  the  equation  can  be  reduced  to  the  form 

Xdx  4-  Ydy  =  0, 

in  which  X  is  a  function  of  x  only,  and  r"  is  a  function  of  y  only. 
Let  the  form  be 

XYdx-\-X'Tdy=^0,  (2) 

in  which  X  and  X  are  functions  of  x  only,  and  T  and  F'  are  func- 
tions of  y  only. 

Dividing  by  XT, 

fF  +  |<?^  =  0,  (3) 

in  which  equation  the  variables  are  separated. 
For  example,  given 

(1  -  xfydx  -  (1  +  y)x^dy  =  0. 
Dividing  by  x^y^ 

as*  y 


DIFFERENTIAL  EQUATIONS.  219 

Hence,  ^_2^  +  cte-^-d2,  =  0. 

ar        a;  y 

Integrating,       2  log  ic  +  a;  —  log  2/  —  2/  =  C'. 

X 

PROBLEMS. 

1.  {l-y)dx^{l-\-x)dy  =  0.  ^ns.  log  (1  +  a;)  -  log  (1  -  y)  =  C. 

2.  (1  H-  2/^)  t^a;  —  x^dy  =  0.  J.ns.  2  a;^  —  arc  tan  y  =  C. 

3.  (l-\-x)ydx-^(l —y)xdy  =  0.        Ans.  log  (xy) -\- x  —  y  =  G. 

4.  dy  -\-y  tan  a;  da;  =  0.  -4ns.  log  3/  —  log  cos  x—  G. 

dx     (1  4-  x^)  xy 

^dy^l±t^  ^^^  x±^, 

dx     l-\-x'  ^     1-Gx 

7.   sina;cos2/(iaj  — cosa;sin2/c?y  =  0.      Ans.  cos  2/ =  O cos  a;. 

8.  Helmholtz's  equation  for  the  strength  of  an  electric  current  G  at 

W      T  dO 

the  time  ^  is  O  = ,  in  which  E,  R  and  L  are  given  constants. 

R     R  dt 
Find  the  value  of  G,  determining  the  constant  of  integration  by  the 

condition  that  its  initial  value  shall  be  zero. 

9.  The  equation  showing  the  strength  of  current  i  for  the  time  t 

dG 
after  source  of  E.M.F.  is  removed,  is  RG  =  —  L—  {R  and  L  being 

constants).     Find  the  value  of  G. 

Ans.  G  =  le  ^,  in  which  /=  current  when  ^  =  0. 

10.   The  differential  equation  of  the  current  of  discharge  from  a 

condenser  of  capacity  O  in  a  circuit  of  resistance  i2  is  -^  =  — — • 

i       GR 

t 

Find  i,  if  the  initial  current  is  Iq.  Ans.  i  =  IqB^', 


Art.  136.     Homogeneous  Differential  Equations. 

The  differential  equation  Mdx  -\-  Ndy  =  0  is  said  to  be  homo- 
geneous when  M  and  N  are  homogeneous  functions  of  x  and  y  of  the 
same  degree. 


220  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

If  the  equation  is  written  in  the  form 

dx~     N' 

*-!  y 

the  second  member  is  seen  to  be  a  function  of  — 

X 

•  y  J 

If,  now,  V  be  substituted  for  -,  ^  -  nr-*  ^   <^,  •    ^ '^^  ^ 

y  = 

and  the  equation  becomes 


«<t^ 


-I  dw        dv  ,  —    =    ^^  f  >c  - 

y  =  vx,  and  -^  =  x \-v,  <<>  ^^ 

dx        dx 


dv  ,  /./  N 

in  which  the  variables  can  be  separated,  giving 

dx_      dv 

For  example,  given  the  homogeneous  equation, 


2  .  ^<^y        dy 

dx  dx 


Substituting  y  =  vx, 


2  0  ,  a?{xdv  +  vdx)  _  vx^{xdv  -\-  vdx) 
dx  dx 

,                                                dv  ,  dx       -, 
whence  1 =  av. 

V  X 

y 

Integrating  and  substituting  t*  = "-, 

y  y 

log-  +  loga;  =  -  +  c. 

y 

Therefore  log  y  —  c  =  -, 

or  Cie^=2/.(logCi  =  c). 


PROBLEMS. 

1.  {x-2y)dx  +  ydy==0.  Ans.  log  (x  -  y)  -  ^—-^  =  c. 

2.  (2y/xy  —  x)  dy  -\-ydx=0.  Ans.  y  =  ce^i. 


to  m  a  i.uuci-.4Ki  <v  ■  . 


ROe<.EM3 


DIFFERENTIAL  EQUATIONS.  221 


3.   a^dy  —  y^dx  —  xydx  =  0.  Ans.  log  x-\--=  c. 


y 


4.  xdy  —  ydx  =  dxVx^  —  y^.  Ans.  log x  =  arc  sin  -  +  c. 

5.  (8y-\-10x)dx  +  (5y-\-7x)dy  =  0,      Ans.  (y -^ xf  (y -\- 2 xf  =  c. 

6.  (oc^  +  y^) dx  —  2  xy dy  =  0.  Ans.  ci^  —  y^  =  ex. 

7.  Find  the  curve  in  which,  the  subtangent  is  equal  to  the  sum  of 
the  abscissa  and  ordinate  at  any  point  of  the  curve. 

From  Art.  70  (3),  it  is  seen  that  the  differential  equation  is 

dx 

-y  —  ^x  +  y; 
dy 

whence  y^  -\-2xy  =  c,  a  hyperbola. 

Art.  137.     The  Form  (ax  -i- by  +  c)dx -\- (a'x  +  b'y  +  c') dy  =  0. 

The  equation  Mdx  +  Mly  =  0  can  always  be  solved  when  M  and  N" 
are  functions  of  x  and  y  of  the  first  degree,  or  having  the  form 

(ax  +  by  +  c)  da;  +  (a'x  +  b'y  +  c')  dy  =  0.  (1) 

Assuming  x  =  x'  -^h  and  y  =  y'  -\-k,  and  substituting  in  (1), 

(ax'  +  by'  J^ah-\-bk-\-c)  dx'  +  (a'x'  +  b'y'  +  a'h  +  b'k  +  c')dy'  =  0.      (2) 

In  order  that  (2)  may  be  homogeneous, 

ah  -\-  bk  -\-  c  =  0,  and  a'h  -{- b'k -{- c'  =  0, 


giving  h  =  — -,  and  Jc==  — 

a'b  —  ab'  a'b  —  a6' 

Equation  (2)  now  becomes 

(ax'  +  by')  dx'  +  (a'x'  +  b'y')  dy'  =  0, 

a  homogeneous  equation,  and  the  variables  can  be  separated  as  in  the 

preceding  article. 

a'      b' 
This  method  evidently  fails  when  a'b  =  ab' ;  that  is,  when  —  =  — 

a'      b'  ^ 

In  this  case  put  —  =  —  =  m,  a'  =  ma,  b'  =  mb. 
a      b 

Equation  (1)  now  takes  the  form 

(ax  -j-by  -\-  c)dx  +  [m (ax  +  by)  +  c'Jdy  =  0.  (3) 


222  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

Assuming  ax -{- by  =  z,  dy  =  ^^~^^^, 

b 
and  substituting  in  (3), 

whence  [i)(z-\-c)  —  a  (mz  +  c')]  dx  +  (mz  +  c')  dz  =  0, 

and  do^H- (^^  +  <^')dz ^^ 

b{z  -\-  c)  ~  a(mz  +  c') 

and  the  variables  are  separated. 

PROBLEMS. 

1.  {l-\-x-]-y)dx-{-(l-^2x-\-3y)dy  =  0. 
Assuming  x  =  x'  -^h,  and  yz=y'-\-Jc,     ' . 

{l^x'-{-h-hy'-hk)dx'-^(l-\-2x'  +  2h-{-3y^  +  S7d)dy'  =  0, 
in  which  ^i  -|-  A;  +  1  =  0,  and  2^4-3A;  +  l  =  0, 

giving  h  =  —  2,  and  7c  =  1. 

The  equation  now  becomes 

(x'  +  y')  dx'  +  (2  a;'  +  3  y')  dy'  =  0, 
which  is  homogeneous  and  can  be  solved  by  Art.  136. 

2.  -^  =  ax -\- by  -\- c.  Ans.  abx  +  b^y  -{- a -{- be  =  Ce'". 
dx 

3.  (2x  +  y-\-l)dx-{-(4:X-}-2y-l)dy  =  0. 

Ans.  x-\-2y-\-  log  (2x  +  y  —  l)=  C, 

4.  (2x-y-[-l)dx-\-{2y-x-l)dy  =  0. 

Ans.  x^  —  xy-\-y^-\-x  —  y=  C. 

Art.  138.     The  Linear  Equation  of  the  First  Order. 

The  equation  of  the  form 

|  +  P.=  Q,  (1) 

in  which  P  and  Q  are  functions  of  x  only,  is  called  a  linear  equation 
because  it  is  of  the  first  degree  with  respect  to  y  and  its  derivative. 


DIFFERENTIAL   EQUATIONS.  223 

This  linear  equation  admits  of  a  general  solution.  As  tlie  second 
member  is  a  function  of  x  only,  an  integrating  factor  of  the  first 
member  will  be  an  integrating  factor  of  the  equation  if  it  is  a  function 
of  X  only.  To  find  such  a  factor,  put  y  =  Xz,  in  which  X  is  an  arbi- 
trary function  of  x,  and  2  is  a  new  variable.  Then  dy  =  Xdz  +  zdX, 
which  reduces  (1)  to 

Xdz  -^zdX-\-  PXz  dx  =  Qdx.  (2) 

Assume  zdX=  Qdx,  (3) 

then  (2)  becomes  Xdz  +  PXz  dx  —  0; 

whence  —  =  —  Pdx ;  hence  log  z  =  —  I  Pdx,  and  z  =  e~^^'^. 

Substituting  this  value  of  z  in  (3), 

e-^p^aX  =  Qdx,  or  dX  =  e^^'^'Qdx. 

Therefore  X=  Ce^^'^Qdx  +  c,  or  ?/  =  Xz  =  e'^^'^i  Ce^^'^Qdx  -f  cV 

PROBLEMS. 
-       ,         yxdx  a       , 

This  linear  equation  for  y,  put  in  the  general  form,  is 

dy        yx     _     a 

dx      1  +  a^  ~  1+a^' 

which   Cpdx  =  -  f_^^  =  log       ^       .    - 

Hence  the  integrating  factor  is 

Ji-*.  =  eiog  (i+x^)-i  =  (14.  a^-i, 

fe^^^'Qdx  =  f^Cl+L^TI  ^^  ^       ax      ^  ^ 

Therefore  y  =  (1  -f  a;") ^ f' — —  +  c]=  ax-\-c(l  +  a^K 

\(1  +  x'y       1 


m 


224  DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 

2.    x'^+(l-2x)y  =  x'.  Ans.  y  =z  x"  (1  +  ce^). 


*3.    x^  —  ay  =  x-\-l.  Ans.  y  =  cx°- ^ 

dx  a  —  X     a 


4.    Va'  +  ar^- Va2  +  a;2^=a;  +  2/. 


x-\-  Va^+  ^ 

5.    —  cos  a;  H- V  sin  a;  =  1.  ^ws.  v  =  sin  a;  +  c  cos  a?, 

da; 


6.  The  differential  equation  of  electromotive  forces  in  a  circuit  of 
resistance  R  and  induction  L,  when  the  impressed  E.  M.  F.  is  a  sinu- 
soid given  by  the  equation  e  =  E  sinp^,  is  E  sin  pi  =  Iti  -\-  L—-  Solve 
the  equation  for  i,  i  and  t  being  the  only  variables. 


Ans.  i  —  ce^ 


A ==^  sin  (pt  —  <i),  in  which  d)  =  tan~^-^- 


Art.  139.     Extension  of  the  Linear  Equation. 

The  more  general  form,      ^-\-  Py=  Q^T?  (1) 

dx 

in  which  P  and  Q  are  functions  of  x  only,  is  readily  put  in  the  form  of 
the  linear  equation  of  the  preceding  article. 

Dividing  (1)  by  2r,        ^„|  +  ^.=  «•  (2) 

Assume 

n 

z  =  2/""+^ ;  whence  2/"  =  2  "-^,  2/""^  =  2J~^  and  dz  =  —  (n—l) y'^'dy. 


DIFFERENTIAL  EQUATIONS.  225 

Substituting  in  (2)  and  reducing, 

£^(n_l)P2  =  -(n-l)Q, 
which  is  the  form  of  the  linear  equation  of  the  preceding  article. 

PROBLEMS. 

1.  dy  -^  ydx=  xy^dx. 

Dividing  by  y^dx, 

y^dx     y^ 
Assume  z  =  y~'. 

Substituting  and  reducing, 

^-22  =  -2aj. 
dx 

By  the  preceding  article, 

I  Pdx  =  —  2x,  and  x 

z  =  -2e^  fe-^xdx  =  e^  (xe-^  -f-  ^g-^*  +  O). 
Therefore  l  =  f  (Ce^  +  J  +  »). 

2.  ^  =  a^f-xy.  ^ns.  i  =«*  +  !  + Ce**. 
dx        ^         ^  2/' 

3.  Sy'^-af  =  x  +  l,  ^ws.  2/»  =  Ce"  -  ^ii  -  i. 

dx  a         o? 


4.    ^  (a^2/»  +  ajy)  =  1.  Ans,  x 


dx  ^ 


Art.  140.     Exact  Differential  Equations. 

The  equation  Mdx  +  Ndy  =  0  (1) 

is  an  exact  differential  equation,  when 

dM    djsr 


dy       dx  (  ) 


226  DIFFERENTIAL  AND   INTEGRAL   CALCULUS. 

When  the  condition  in  (2)  is  fulfilled,  the  integral  may  be  obtained 
iDy  finding  I  Mdx,  regarding  y  as  constant  and  adding  an  arbitrary  func- 
tion of  y. 

^he  undetermined  function  of  y  may  be  found  by  the  condition  that 
the  differential  of  the  result  just  obtained  regarding  x  as  constant  must 
equal  Ndy,  that  is, 

dyJ  dy 

from  which  f{y)  may  be  obtained.     (See  Art.  49.) 


PROBLEMS. 


1.   ^  +  (2y-^\dy  =  (^. 

y     \       y  J 


dy  2/^'  dx  y^^ 


condition  (2)  is  fulfilled,  and  the  equation  is  exact.      I  Mdx,  treating  y 


as  constant  and  adding  f(y),  is  -  -\-f(y). 


Now 

d 
dy 

■(iy-W"-'-}' 

whence 

-^^— ? 

and 
hence 

,          4flJl.2„ 
dy 

Ay)  =f- 

Therefore 

^+f=c. 

y 

2.  (ioxy--'f)dx-^(Sa^-2xy)dy  =  0.  Ans.  S  oc^y  -  y^x  =  0. 

3.  x(x-\-2y)dx-^(o(^-y^)dy  =  0.  Ans.  a^  +  Sx^y  -  f  =  0. 

4.  {x'-4:Xy-2y^dx-{-(y^-4:xy-2a^dy  =  0. 

Arts,  a^  —  6x^y  —  6  xy^  +  2/^  =  0. 

5.  xdx-\-ydy-\-^^y~  -^f^  =  0.  ^ws.  ar^  i- 2/' -  2  arc  tan  ^  =  C. 

6.  e{Q^-\-y^^2x)dx^2yedy  =  ^.  Ans.  e' (x^ -{- y^  =  C. 


DIFFERENTIAL  EQUATIONS.  227 

Art.  141.     Factors  Necessary  to  make  Differential  Equations 

Exact. 

When  Mdx  +  Ndy  is  not  an  exact  differential,  it  may  often  be  trans- 
formed into  an  exact  differential  by  the  introduction  of  a  factor  con- 
taining X  OT  y  or  both.  This  factor,  which  converts  a  given  differential 
equation  into  an  exact  differential  equation,  is  called  an  integrating 
factor. 

I.     When  Mdx  +  Ndy  is  homogeneous. 

Mdx  +  Ndy  =  \  [{Mx  +  Ny)  {^  +  ^j^  (Mx  -  Ny)  f—  -  — )  1 

=  h[(Mx-hm(i^og(xy)-\-(Mx-Ny)d]og''[    (1) 
TT  Mdx  -\-  Ndy      ,   , ,      /     x   ,    i  Mx  —  Ny  , ,      x 

=  id[log.  +  log/]  +  ifcf*;,when.  =  ?. 

Mx  -f  Ny  V  y 

(2) 
When  M  and  N  are  homogeneous, 

Mx-Ny^... 
Mx  +  Ny     -^  ^  ^' 

and  the  second  member  of  (2)  is  an  exact  differential. 

Therefore is  an  integrating  factor. 

Mx  +  Ny 

This  method  fails  when  Mx  -{-  Ny  =  0,  but  iti  this  case  Mx  =  —  Ny. 
Dividing  the  first  term  of  Mdx  -f-  Ndy  =  0  by  3fx,  and  the  second  term 

by  its  equal,  —  Nyj 

dx _dy _  r. 
X       y 

Therefore  y  =  Cx. 

For  example,  given  (xy  -{-y^dx—  (x^  —  xy)  dy  =  0. 

1  1 


Here 


Mx  -\-  Ny      2  xy^ 


228  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Multiplying  by  this  factor, 

The  condition  of  integrability  for  an  exact  differential  is  now  ful- 
filled. 

Therefore  -  +  log  {xy)  =  G. 

II.    The  form,  /i  {xy)  ydx+f^  (xy)  xdy  =  0. 

By  a  method  similar  to  that  of  I.,  it  may  be  shown  that 


Mx  —  Ny 

is  an  integrating  factor.  This  fails  when  Mx  —  Ny  =  0,  but  it  can  then 
be  shown,  as  in  the  corresponding  case  of  I.,  that  the  solution  now  is 
xy=  C. 

But  another  method  of  solution  by  the  separation  of  the  variables 
may  be  used.     For  example,  given 

(x^y^  -\-  xy)  ydx-{-  (x^y^  —  l)xdy  =  0.  (3) 

Assume  xy  =  v, 

then  ^^^^v)'l^dx+(v^-l)x(^^-'^^  =  0; 

whence  ^  =  _f!LzilV^,  (4) 

X  \    V     J 

and  the  variables  are  separated. 

Integrating  (4),  and  substituting  v  —  xy, 

y  =  cev. 

III.  To  determine,  the  factor  necessary  to  render  Mdx  +  Ndy  exact, 
when  that  factor  is  a  function  of  one  variable  only. 

Assume  X,  a  function  of  x  only,  to  be  the  required  factor;  then 
XMdx  +  XNdy  is  an  exact  differential. 

Hence  ^  {XM)  ^  ^  {XN) -, 

dy  dx 

and  since  -r-  =  0, 

dy 


DIFFERENTIAL  EQUATIONS.  229 

By  dx  dx  ^ 

..       o  dX      1  fdM     dN\.  .K\ 

therefore  _  =  _^____j^..  (5) 

The  first  member   of  (5)  does   not  contain  y,   hence  the   second 
member  must  be  independent  of  y  also,  or 

^(f-f)-=/<")- 

Integrating  (5),         logX=  lf(x)dx, 

therefore  X  =  e^^^'^''^ 

Similarly,  if  Y,  a  function  of  y  only,  is  the  original  factor, 

YMdx  +  YNdy 

is  an  exact  differential,  and  ^ 

Y  =  e^'^^^^^^. 
For  example,  given  {x^  -\-  y"^  -{-  2  x)  dx  -\-  2  ydy  =  0, 
Assume  X  to  be  the  integrating  factor. 

N\dy        dx  J 
log  X  =  I  dx  =  X,  and  X  =  e*. 
Multiplying  the  given  equation  by  e% 

e^(x^  +  2/2  -f-  2  ») die  +  2  e'ydy  =  0. 

The  condition  of  integrability  for  an  exact  differential  is  now  ful- 
filled, and  integrating  as  in  Art.  140, 


230  DIFFERENTIAL   AND  INTEGRAL  CALCULUS. 

PROBLEMS. 

1.  x(a^-^Sy^)dx  +  y(y^-\- Safety  =  0.      Ans.  x' +  6x^y' +  y' =  c. 

2.  ydy-\-xdx-\ „~\ — •  Ans.  — ^tX  _)_  arc  tan  -  =  c. 

ar  +  2/  2  X 

3.  (pi?  -\-  y'^)dx  —  2  xy  dy  =  0.  Ans.  x^  —  y^  =  ex. 

4.  (x^  -{-2xy  —  y^dx=x^  —  2xy  —  y^dy.   Ans.  x^  +  ?/-  =  r (ic  +  y). 

5.  (x  -f  v)^-    =  ctl  Ans.  y  —  a  arc  tan      '  ^  =  c. 

dx'  a 

■6 .  x^dx  +  (o  .r^;/  -f  2  2/''^)  <^2/  =  0.  ^?is.  a^  +  2  ?/'  =  c  \/ar^  +  y\ 

7 .  (1  +  ic^/) 2/ '^•^  +{1  —  icy)xdy  =  0.  Ans.  x  =  cye^^^. 

•8-  (2/  H"  y^^y)  dx  +  {x-\-  x-\/xy)dy  =  0.  -4m6\  xy  =  c. 

9.  (3  a^  -  2/2) ^  =  2  i»2/.  ^ns.  x'-y^  =  cy\ 

(XX 

10.    2  xy  dy  =  (a^  -[- y^  dx.  Ans.  ':/?  —  y^  —  ex. 

Art.  142.     First  Order  and  Degree  with  Three  Variables. 

The  general  form  of  the  differential  equation  of  the  first  order  and 
degree  between  three  variables  is 

Pdx  -f  Qdy  +  Rdz  =  0,         .  (1) 

in  which  P,  Q  and  B  are  functions  of  x,  y  and  z. 

Such  an  equation  sometimes  admits  of  solution  by  the  separation  of 
\the  variables. 

To  obtain  the  condition  of  integrability,  represent  the  function  by 

v=f(x,y,z)  =  c',  (2) 

whence  ^  dx -{•  ^  dy -\- ^  dz  =^  0.  (3) 

ox  dy  oz 

Comparing  (1)  and  (3),  it  is  seen  that 

P,  Q  and   B  are  proportional  to  -^,    ~  and  -^' 
^  ^  ^    ^  dx     dy  Bz 


DIFFERENTIAL   EQUATIONS.  231 

To  obtain  the  required  relation  between  P,  Q  and  B,  assume  the 

factor  u  such  that 

dv 


«^=i' 

(4) 

' 

»«=& 

(5) 

and 

dz 

(6) 

From  (4)  and  (5), 

hence 

ay       dy           dx          dx 

Similarly,  »(f-|>«|"-Q|.  _(«) 


and 


JdR_ap\pdu_j^du_  (9) 

\da;       62;  y         62  dx 


Multiplying  equations  (7),  (8)  and  (9)  by  72,  P  and  Q,  respectively, 
and  adding, 

'■(S-f)*<f-f)-'=(^-S)=»-     <'»' 

Equation  (10)  is  the  required  condition  of  integrability.  When 
this  condition  is  fulfilled,  equation  (1)  may  be-  integrated  by  regarding 
one  of  the  variables,  aJ,  y  or  z,  as  constant,  and  omitting  the  correspond- 
ing term,  Pdx,  Qdy  or  Pdz. 

Thus  omitting  Rdz,  integrating  Pdx  -\-  Qdy  =  0,  regarding  z  as  con- 
stant and  introducing  /  (z)  as  the  constant  of  integration,  the  integral 
is  obtained  so  far  as  it  depends  upon  x  and  y.  Finally,  by  comparing 
the  total  differential  of  this  result  with  equation  (1),  df(z)  is  found  in 
terms  of  2;  and  dz,  and  then  by  integration  the  value  of  f(z). 

When  certain  terms  of  the  equation  form  an  exact  differential,  the 
remaining  terms  must  also  be  exact.     It  follows  that  if  one  of  the  vari- 


232  DIFFERENTIAL   AND  IN^TEGRAL   CALCULUS. 

ables,  say  Zj  can  be  completely  separated  from  the  other  two,  so  that  in 
equation  (1)  B  becomes  a  function  of  z  only,  and  P  and  Q  functions  of 
X  and  y  only,  the  terms  Pdx  +  Qdy  must  be  thus  rendered  exact  if  the 
equation  is  integrable. 

For  example,  given  zy  dx  —  zxdy  —  y'^dz  =  0.  Dividing  by  y\  which 
separates  z  from  x  and  y,  and  puts  it  in  the  exact  form, 

ydx  —  xdy     dz  _  ^ 
of  which  the  integral  i^  x  =  y  log  cz. 

PROBLEMS. 

1.  (x  —  Z  y  —  z)  dx  +  (2  y  —  Z  x)  dy  -\-  (z  —  X)  dz  =  0. 

Ans.  x^-i-2y^-6xy-2xz-\-z^=C. 

2.  yz^dx  —  z^dy  —  e'dz  =  0.  Ans.  yz  =  €"(14-  cz). 

3.  yzdx-\-zxdy  -{-  yx  dz  =  0.  Ans.  xyz  =  G. 

4.  {ydx-\-x dy)(a  -{-z)  =  xy  dz.  Ans.  xy  =  c{a-\-  z). 

5.  The  increase  in  energy  of  a  magnetic  field  caused  by  the  inde- 
pendent increments  di^  and  di^,  of  current  in  two  coils  of  self-induction 
Li  and  L^  and  mutual  induction  fx  is  dW  =  Liiidii-\-  L2i2di2-\-  ixiidii 
+  fiiidii.     Find  the  energy  of  the  field. 

Ans.    W=^f  +  ^  +  f.Hi2. 

Art.  143.     First  Order  and  Second  Degree. 

The  general  form  of  a  differential  equation  of  the  first  order  and 
second  degree  is 

da^  dx  '  :  ^^ 

in  which  M  and  N  are  functions  of  x  and  y.  The  direct  differential 
obtained  from  the  primitive  contains  only  the  first  power  of  -^,  and 

hence  cannot  be  identical  with  (1).  But  if  the  primitive  is  supposed 
to  contain  the  first  and  second  powers  of  a  constant  c,  and  is  solved 
with  reference  to  c,  there  will  result  two  values  of  c,  from  each  of  which 
c  will  disappear  on  differentiation ;  and  each  of  these  resulting  differ- 


DIFFERENTIAL  EQUATIONS.  233 

ential  equations  will  contain  only  the  first  power  of  --^,  each  being  a 
factor  of  (1).     Hence  the  product  of  the  two  equations  will  give  (1). 
Therefore  the  given  equation  is  solved  as  a  quadratic  in  -^,  all  the 

terms  are  transposed  to  the  first  member,  and  it  is  resolved  into  two 
factors  of  the  first  order  and  degree.  Each  of  these  factors  is  then 
placed  separately  equal  to  zero  and  integrated,  using  the  same  arbitrary 
constant  in  each.  When  all  the  terms  in  each  of  these  results  are 
transposed  to  the  first  member,  the  product  of  these  first  members 
placed  equal  to  zero  will  be  the  complete  primitive. 

For  example,  given     y-^-\-2x^  —  y  —  0.  (1) 

dor  ax 

Solving  for  -^j 
dx 


^^_2_^V^T2,  and  ^y  =     ^      Va?^  +  /. 
dx         y  2/       '  dx         y  2/        ' 

whence         dx  =  +  ^^^±l^,     and  do.  =  -  ^^^±1^.  ^        (2) 

Integrating  (2), 

a;  =  -f-  Var^  +  2/^  +  c,  and    x  =  —  -s/x^  -\-y^  4-  c. 


Therefore      {x  —  c  —y/oi?  +  y'^{x  —  c  +  VV  +  ^  =  0, 
or  2/^  =  c^  —  2  ex. 


PROBLEMS. 

1.  -^  =  oic.  Ans.  (y  ~  c)^  =  |  ax^. 
dor 

2.  ^-5^  +  6  =  0.  Ans.  (2/-2x  +  c)(2/-3a;  +  c)  =  0. 
dor       dx 

3.  x'^4-Sxy^-{-2f  =  0,        Ans.  (xy -\- c)  (a^y -^  c)  =  0. 

dor  dx 

4.  (a;2_^  1)^  =  1.  Ans.  (^6^^-20X6^  =  1. 

dor 

5.  ^(^^y\  =  x(x-\-y).  Ans.  (x'-2y  +  c)l6'(x-\-y-l)-hc]. 
dx\dx       J 


234 


DIFFERENTIAL   AND  INTEGRAL   CALCULUS. 


Art.  144.     Differential  Equations  of  the  Second  Order. 

J.   Equations  involving  x  arid  — ^  only. 

dar 

The  equation,  if  possible,  is  put  in  the  form 


(1) 


dx 


(2) 


in  which  X  is  a  function  of  x  only. 
Integrating  (1), 

Integrating  (2),  y=  X^-hC^x-^-  O^.  (3) 

In  (2)  and  (3),  Xi  and  Xg  are  functions  of  x  only,  and  Ci  and  C2  are 
arbitrary  constants. 
For  example,  given 


Then 


dx^-''''' 
d^y 


dx 


ax*dx, 


dy^ax^^ 
dx       0 

,       ax^dx  ,  ^  ■, 
dy  =  — - — h  Cidx, 


y  =  ^+ax-^C^ 

d^y 
II.   Equations  involving  y  and  — ^  only. 

aXi 

The  equation,  if  possible,  is  put  in  the  form 


d^~^' 


in  which  F  is  a  function  of  y  only. 


Multiplying  (4)  by  2  -^, 
ax 


2^^  =  2Y 


dx  dx^ 


dx' 


whence 


2^d(^)  =  2Tdy. 
dx    \dxj 


(4) 


(5) 


DIFFERENTIAL  EQUATIONS.  235 

Integrating  (5), 

Separating  the  variables  and  integrating, 


yJ2JTdy-{-C, 


For  example,  given  — ^  =  a^y. 

Then  2^^  =  2a'y% 

dx  dor  dx 


Vay  +  c, 


III.    Equations  not  involving  y  directly. 

The  equation  will  be  of  the  form  F  (x,  %  ^\  =  0. 

\      dx  dx^J 

Assume  ^  =  .,  then  ^  =  ^. 
dx  dx^     dx 

Making  these  substitutions,  an  equation  of  the  first  order  between 

z  and  X  is  obtained. 


For  example,  given  g  +  ^rV[l+(|)]=0- 

Assume  -^  =  z, 

dx       ' 


I       then  ^^ =  —  '^cix. 


dz ^_25 

~  ,2 


V(l  +  zy  a' 

Integrating, 

-7^  =  -  "l  +  (?,  =  ^^  when  C  =  4; 


^nce 

z  = 

.dy  _ 
'  dx~ 

<^ 

-a? 

Va^- 

(c^-: 

Tff 

therefore 

y-- 

r  (I 

dx 

iV. 

Equations 

not 

involving 

X  directly. 

Assume 

dy 
dx^ 

=  z, 

n 

d'y_ 

dx' 

_dz  _ 
''  dx^ 

_dzdy  _ 
dydx 

dy 

Making  these  substitutions,  the  independent  variable  is  changed 
from  x  to  2/,  and  an  equation  of  the  first  order  between  z  and  y  is 
obtained. 

For  example,  given         ^  -  a  (^  =  y.  (6) 

dar        \dxj  ' 

Substituting  ^  =  «,  and  ^  =  z  — , 
dx  dx^        dy 

^%-a.^  =  y.  (7) 

Assume  z^=-2v,  then  zdz  —  dv,  and  substituting  in  (7), 

dv  —  2avdy  =  y  dy.  (8) 

Equation  (8)  is  a  linear  equation  of  the  first  order  and  degree  and 
is  therefore  integrable. 

PROBLEMS. 

Ans.  y  =  x^  -{■  (J^x  +  C^. 


1. 

2. 

3. 

d?y_    1    . 

4. 

dxr     dx 

Ans.  X  — ^—  log        ^       ^- h  Cg. 

V2^       VOiV^  +  l  +  l 

Ans,  2^  =  Ci  log  ic  +  (72- 


DIFFERENTIAL  EQUATIONS.  237 

5.    (1  -  ar*)  ^  -  x^  =  2.     Ans.  y  =  (arc  sin  xy  +  Ci  arc  sin  a;  +  Og. 

••-(3)'-+©"  ''-¥-«'^+^+<^. 

8.   2,(l-log2,)U  +  (l  +  log2//|J  =  0. 

^715.    log  2/  —  1  = 


Ci«  4-  Ci 
9.2/^- (^=  f  log  i/.    ^Tis.  log  y  =  C,e  +  C^e-'. 


10.    The  equation  of  motion  of  a  particle  ascending  in  the  air  against 
the  action  of  gravity  is 


=-^-Ki)' 


•Find  the  equation  for  the  space  described  by  the  particle  in  terms  of 

dx 
the  time ;  determining  the  constants  of  integration  by  making  —  =  '^ 

when  t  =  0,  and  a;  =  0  when  ^  =  0. 

Ans.  X  =  — -  log  (vk  sin  kgt  +  cos  kgt). 
gk 


APPENDIX, 


>>©4< 


TABLE   OF   INTEGRALS. 

This  table  contains  the  principal  integrals  given  in  this  book,  to- 
gether with  a  few  additional  ones.  The  arrangement  only  partially 
follows  the  order  in  which  the  forms  occur  in  the  book,  as  convenience 
of  reference  is  the  first  consideration. 

I. 

ELEMENTARY  FORMS. 

w. 


1.     i  {du  +  dv —  dw)  =  u-[-v 
2.     iadx=ax.  4.     la— =  a  log 

8.     Cadf{x)  =  af{x).  5.     \ax''dx 


X, 

n+l 


EXPONENTIAL  FORMS. 


6.     j  a'  log  adx=za'.  '^'     \  ^'^^  ~ 


TRIGONOMETRIC  FORMS. 


8.  j  cosa5C?a;=  sinic.  10.     j  sin  a; da;  =  — cos  a;. 

9.  j  sec^a;daj  =  tana;.  11.     j  cosec'^a;da;  =  — cota;. 


238 


APPENDIX.  239 

12.  (  secxta,nxdx=secx.  14.     j  sin  a;  cia;  =  vers  a?. 

13.  j  cosecfl;cota;dcc  =  — cosecjc.       15.     j  cos  a;  die  =  — covers  a;. 

INVERSE   TRIGONOMETRIC    FUNCTIONS. 

/fix                                                          r       dx 
=  arc  sin  x.  20.     I z=:  =  arc  sec  x. 

vnTo^  -^  xVx"  - 1 

17.     f ^^      =  arc  cos  x,  21.     ( --^ =  arc  cosec x. 


dx 
VI  -  x^  ~~  ^^^   ""  ~*  "^       a;Va;^-l 

da; 


18.  r-^  =  arctana;.  22.     f 
J  1  +  ar^  •^ 

19.  r--^=arccotaj.  23.     f--^ 

J      l  +  ar*  -^      ^2x 


-s/2x-x' 
dx 


arc  vers  x. 


=  arc  covers  a;. 


II. 

RATIONAL  ALGEBRAIC   FORMS. 
Expressions  containing  (a  +  hx). 


24.     r_^  =  ^log(a  +  6aj). 
•/  a  +  6a;     b 

»/  a  4-  6a;     6^ 

26.  fv^^-i^  da;  =  V(a  +  a;)(6  +  a;)+(a  -  6)  log  ( VaT«  -f  Vb+~x). 
J  ^0  -\-x 

27.  r\/^^^^  da;  =  V(a  -  a;)(6  +  a;)  +  (a  +  6)  arc  sin \/^i^. 

28.  r-^^  =  i[i(a4-6a;)2-2a(a  +  6a;)+a21og(a  +  6a;)]. 

*/  Ct  -f-  Ox       0 

c/  x{a-\-ox)         a  X 

J  xr{a  +  bx)         ax     a?  x 


240 

J  (a 
32 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 
dx  1 


32.      f- 

"     J  (a 

J  (a 


(a  +  bxf         6  (a  -I-  bx) 
xdx 


(a  +  bx) 

(a  +  bxf      b' 
dx 


i  flog  (a +  60^)4- -^1- 
¥[_  a  +  bxj 

a-\-bx  —  2alog(a-\-  bx) — |. 

a  +  bxJ 


34 


r__d^L_^__l___l.fa^bx 


'  (a  +  6a;)^     a(a-{-  bx)      a^       \     x 


35. 


»/  a 


Expressions  containing  (a  4-  bx^. 

dx         1.x 
„ =  -  arc  tan  — 

^  -^xr     a  a 


40 


41 


36.  r-^=  1  iog^+^. 

J  a^  —  xf-     2a       a  —  a? 
37 .     f    ^^   „  =  -^  arc  tan  a;  \/-,  when  a  >  0,  and  ?  >  0. 

38-     f-T^lo  =  A  logf^^\  ^l^en  a  >  0,  and  ^  >  0. 
J  (T  —  6 V     2  a6        \a  —  6a;y 

J  or  —  or  ^x  +  a 

Ja  +  6a^     26     ^V     ^/ 

/'  a^dx        X        fa        .  fb 

J  x^(a  4-  6ar)          aic      ^f  a^  ^f  a 

43.     f — ^^ = E _j i —  arc  tanaj\/-. 


44 


45 


'   J  (a 


+ 


2m-l 


c?a; 


+  5a^"*+^     2  ma  (a  4-  ^a^)"*        2  ma    J  (a  +  bx^ 


4- 


fta^^'+i     2  m6  (a  +  6x2)m     2  m6  J  (a  +  bx^) 


-f- 
mb  J  (a 


dx 


APPENDIX.  241 

Expressions  involving  (a-^baf). 


46 


.     i  OCT  (a  +  bafydx 

^m-«+i  ^(j  _^  bx^'Y^^  —  (m  —  n  +  1)  a  Cx""""  (a  +  bx^dx 
~~  6  (wp  +  m  + 1) 

47 .  n»-'"  (a  +  bx^y  dx 

.^-m+i  ((J  4.  5a;'*)p+^  +  6  (m  —  np  —  n  —  1)  j  a;-"'+"(a  +  bx'^ydx 
—  a(m  —  1) 

48.  Cx'^ia-^-bxydx 

a;"'+i  (a  +  6a;''y  +  anp  |  a?"*(a  +  6a;'*)^'"^da; 
np  -f-  m  +  1 

49.  Cx"^  (a -{-bi '*)-'' dx 

ar+'\a  4-  60;")-*'+^  —  (m  +  n  +  1  —  np)  j  af  (a  +  6a5")-»^^daj 
~"  an{p  —  l) 

III. 

IRRATIONAL  ALGEBRAIC  FUNCTIONS. 


50.     CVa  +  bx  dx=—  V(a  +  bxf. 
J  ob 

51     f-^^=  =  ?V^T6^. 

'   J  ^a-\-bx      b 


Expressions  containing  Va  +  bx, 
2 


52.   J  a;VaH- toda;  — 


2(2a-3&a;)V(a  +  &a?y 


''•     ■      -^75^-  36^ 


^_2(2a-Mv^T6^. 


J        X  V  Va  +  &ic  +  Va/ 


242 


55. 


56v 


57. 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 
r       ^^        =-1-  logf^^^^\  when  a>0. 


6a;  15  6=^ 


a;^  Va  -|-  6a; 


VaH-  6a; 
aa; 


6     ,      f^a  -\-hx—  Va 


2 Vo^        VVa  H-  6a;  +  Va, 


i)- 


Expressions  containing  VoM-^- 

58.  rV^^T^c?a;  =  |V^M^  +  ^'log(a;  +  VaM^. 

59.  f— ^=:  =  log(a;+ V^M^. 
-^  Va^  +  a;2 

60.  f   ^^^    =V^^T^. 

*^  Va^  +  a;^ 

61.  J^^dx  =  V^+^  -  a  log  («  +  vj+^). 

62.  f       '^^        =llog — g 


63 


64. 


oi^dx         X 


,.__  =  I  VSq^  - 1  log  (0!  +  Va^  +  0^). 

Va^  4-  ar     -^  ^ 


65 


66 


•^8  8 


/da;       _  a; 

(a2  4-  a;2)|      a^  V^+^ 

67.     ^{a"-  +  af)^  da;  =  I  (2  ar'  +  5  a^)  V^T^  +  ^  log  (a;  +  VoM^. 
*/  8  8 

68      C        ^^         ^      VoM^  ,     1    w^^  +  V^H^ 
'   -'a;«V^M^  2aV    "^  2  a«     ^^^  a;  7 


APPENDIX. 


243 


69.     f- 


dx 


VoM-^  dx         Vci^  +  x^ 


70 


.p 


a^ 


+  log  a;  +  Va^  +  a^. 


71. 
72. 
73. 

r   74. 

75. 
76. 


Expressions  containing  Va^  —  a^. 
j  Va^—  a^  c?aj  =  |-  [a;Va^  — a^  +  a^  arc  sin  -  ]• 

/da;  .    a; 

-^=r  =  arc  sm  — 
Va^-x"  « 

j  ic  ^G?  —  y?  dx  =  —  \  ^{p?  —  ar*)^. 
f       ^^        =  i  log  f ^ ^^. 


77. 


78. 


79. 


80. 


81. 


82. 


J  X  X 

fx"  Va-  -  xFdx  =  -  -V(a2-ar^)3  +  ^YaVa^-a^  +  a^  arc  sin  ^\ 

/dx 
a^  Va^  —  a^ 


arc  sin  — 
2  a 


^/a'-x" 


V«^-^^^^_Va^-^_arcsin^. 


f  V(a2  -  a^)3  da;  =  J  ["a;  V(a2  -  ar^«  +  ^  y/oT^  +  ?|-'  arc  sin  -l 


da;         _  a? 

V(a^  —  x^Y     a-  Va^  —  ar^ 


244  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

83.     I —  = —  —  arcsm-' 


Expressions  containing  Var^  —  a?. 

85.  j  VaJ^  —  (^dx  =-\\x Va?^  —  a?  —  o? log {x  +  Vaj^  —  a^)]. 

86.  r      ^^       =log(;a;4-V^^"^=^^. 
•^  Va?^  —  a* 


xdx 


87.  r-4^^=vs^^^^. 


88 .  fa;  Var^  -  a'dx  =  i  V(a^  -  a^)^ 

89.  fVa^  -  a'  ^^  ^  Va^  -  a'  -  a  arc  cos  -• 
J  X  X 


90.     r ^^        ^- arc  sec 

^  a; 


da;  1  X 

^3=:^^  =  -  arc  sec  — 

Va^  -  a^     a  a 


91.     f-^^  =  ^V^^:=^2^^' log  (a;  H-Va^-a^); 


Va^  —  a^     ^ 

c?a;  Va^  —  a^ 


go         I  

•^ar^Va^-a'  «'^ 

93.     fa^  Va^  -  a^c^a;  =  |(2  a^  -  a')^^^  -  a'  -  -log  (aj+Va^-a^. 
»/  8  8 


/c?a;  Va^  —  o^^  ,     1    „„«  o^«  ^ 

95.   jV{x^  -  a2)3da; 

=  I  ra;V(a;=^-aT  -  ^  V^^'^^^  +  ^  log  (a;  +  V^^^^)]- 


96. 


/ 


APPENDIX.  246 

dx  X 


V(ar^  -  ay         aWx"  -  a» 


Jq?(1qc  X 


98       ■         ^^ 


99.     ((x"  -  a^)^  da;  =  |(2  a^  -  5  a=^)  Var^  -  a^  4-  ^  log  (a;  +Var'-a^. 
«/  8  8 


Expressions  containing  V2  aa;  ±  a^. 

.     rV2  aa;  -  afdx  =  ^::i^  V2aa;-ar^  +  ^  arc  ver -. 
J  2  2  a 


01.     I —  =  arc  ver 


^^         =  arc  ver  -. 

V2  aa;  -  ar^  « 


0/5.     I     .  =  =  —  V2  aa;  —  ar^  +  a  arc  ver  -• 


03 


*^  a;^ 


.^^ _  V2  aa;  -  ar^ 


V2  ax  —  Qi?  ax 

04.     fa; V2^a^^^^da;  =  -  3a^  +  fta;-2g' ^2ax-x'  +  ^' arc  ver  ^. 
J  6  2  a 


05 .     rV2a.r-ar^^^  ^  V2aa-ar^  +  a  arc  ver -. 
J  X  a 

^g^     rV2  aa;  -  ^^^^_/2  ax -a^^f, 
*^  a:^  V    Sax'    ^ 

rv,v      r        c^a;  a;  —  a 

07.     I j  = — =• 

^  (2  ax  -  o^^     aV2  ax  -  ar» 

08.  r  ^^^  .=-— ^ — 

*^  (2  ax  -  x^)^     a  V2  ax  -  x* 

09.     r       ^^       =  log  (x -4- a-^-V2  ax +  x^. 
^  V2  ox  +  x2 


246  DIFFERENTIAL  AND  INTEGRAL   CALCULUS 

110.     I —  = ^7- — -^2 aic  —  ar  +  I  a^  arc  ver -. 

-^V2ax-a^  2  a 

lit,     r^^^,^  =  _/^^+6aa;+5a2^V2^^"^r^  +  |a 
*^  V2  ax  —  a^         \3  / 

#^ 

Expressions  containing  Va  +  6aj  ±  cxl 


arc  ver  ?. 


112.     r  ^^     =— log  (2  ca;  4-  6  +  2  V^  VaT6^"+^). 

»^  Va  +  6fl;  4-  car^      Vc 


4c 
6^  —  4  ac 


log  (2  ca;  +  6  4-  2  Vc  VoT^^T^). 


=  —    arc  sm 


c^ 

2car-6 


VP  +  4ac 


114.  r    ^^     =-^ 

*^  Va  4-  6x  —  ca;^      Vc 

115.     I  Va  4-  6a;  —  cMx  —     ^^  ~     Va  4-  6a;  —  car^ 
J  4c 


,  6^  4-  4  ac  .       2  ca;  —  6 

4 — arc  sm 


116. 


/ 


8c2  V6'4-4ac 

a;c?a;  Va  4-  6a;  4-  car^ 


Va  4-  6a;  4-  ca;^  ^ 

— ^r  J-  log  (2  ca;  4-  6  4-  2  Vc  Va  H-  6a;  4-  co^) 
2cLVc 

117.     r   ^      ^^^  ^Va4-6a;4-c.-ff-|^^ 

*/  Va  4-  6a;  4-  car^  V^c     4cV 

36^_^\  r  c^a; 

8  c2     2  c  J  J  Va  +  6a;  +  cy? 

x'^dx  a;"~^  Va  4-  6a;  4-  car^ 


118 


•'^  Va4- 


6a;  4-  ca;^  ^^ 

a;'»-2r?a;  2n  —  1     hC        x'^-Hx  _ 

coi? 


n-1     a  r        a;^-^r?a;  2n-\     h  C        x^'-Hx 

^      '  cJ  Va  4-  6a;  4-  car^         2n        cJ  Va  4-  6a;  4- 


APPENDIX. 


247 


IV. 


TRIGONOMETRIC  AND  TRANSCENDENTAL  FUNCTIONS. 


119. 


120. 


121, 


122. 


123. 


124. 


sin^  xdx  =  ^x  —  ^  sin  2  x. 


j  eos^  X  dx  =  ^  X  -i- 1  sm2  X. 

I  tan  xdx  =  log  sec  x. 

I  Gotxdx  =  log  sin  «. 

I  ^-^  =  log  tan  4-  X. 
J  smx 

/; 


'«*        logtanf^  +  i^). 


cosx 


125 


.     j  cosec  xdx  =  log  tan  ^  a;. 

C       dS  2  .      f/^a-^M 

•     I ;^ ;;  =  —  arc  tan  ( -)- 

J  a  +  6  cos  ^      Va^  —  h^  W^  +  ^/ 


^  tant 


,  when  a>b, 


log 


V6  -f-  a  +  V^  —  a  tan  - 


^^'  ~  ^'        V^T^-Vft-atan^ 


when  a  <  6. 


127. 


128. 


129. 


130. 


131. 


j  a;  sin  a;  da;  =  sin  a;  —  a;  cos  «. 

j  x^  sin  a;  da;  =  2  a;  sin  a;  —  (a;^  —  2)  cos  x. 

j  a;  cos  X  da;  =  cos  a;  -I-  a;  sin  x. 

I  ar*  cos  a;  da;  =  2a;cosa;H-  (ar^  — 2)  sin  a;. 

/sin  a;  -,  a^    ,    x^        oF    , 


248  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

33.  j  arc  sin  a;  da;  =  ccarc  sin  a;  +  VI  —  ar*. 

34*.  I  arc  cos  a;  c?aj  =  a;  arc  cos  x  —  Vl  —  ar*. 

35.  j  arctana;da;  =  a;arctana;  —  •|log(l  ■\-y?). 

36 .  j  arc  cotan  a;  c?aj  =  a;  arc  cot  a;  -|-  ^  log  (1  +  a?^. 

37.  j  arcversa;c?a;=  (a;  —  1)  arc  vers  a; -f- V2  a;  —  ar'. 

38.  I  log  xdx  —  x  log  a?  —  a;. 

/'  dx  11 
=  log  (log  a;)  +  loga;  +  2^  log^aj  +  ^—3-2  log^a;  -♦-  •• 

40.     r_i^  =  log(logaj). 
J  xXo'gx 

xe'^^'dx  =  ~{a'  —  1). 

A  o      C^ax  c\^  «.  /i^     «***  (<*  sin  X  —  COS  a;) 

42 .  I  e"*  sin  a;  aa?  =  — ^^ — ^« 

J  a^  +  l 

C  ay.  J        e'*''  (a  COS  a;  +  sin  a;) 

43.  I  e"*  COS a5C?a;  =  — ^^ — ^ ^' 

44.  Ce^logxdx  =  C^^^-^  f—dx. 
J  a  aJ  X 

45.  fa;«e«*da;  =  ^^  -  -  fx'^-V^da;. 

46.  ra;«loga;c^a;  =  a;-+fi^---i— -1. 

47.  rsin"a;da;  =  -^^^""^"^^^  +  ^^^  fsin^-a^dc.. 
J  n  n     J 


m  +  n  m-\- 

1 I  cos*"  X  sin"^  xdx. 

m  +  n  m 

.n-l 


APPENDIX.  249 

148.  I  cos*^xdx  =  -Gos*^~^xsmx-] — ^^—  I  cos'*~^xdx. 
J  n  n    J 

149.  f COS"* X  sin"  xdx=^  ^^^"'^  ^  ^^^^^"^  ^  +  ^^^^^  rcos"'-2a;siii"a;da; 
»/  m  -\-n  m-\-  nJ 

150.  ftan" xdx  =  *?^!ll^  _  Aan-^ x dx. 

151.  1  a;"'log**a;c?a;  =  -^^^log"a; ^  fcc^log'-^ajc^a;. 

J  m  +  1  m  +  1*/ 

r^!!^  — a;"*+^  ■  m  +  1  r  a^^'da; 

J  log"  X          {n  —  1)  log""^  ic      n  —  lJ  log""^  a? 

153.     ra"^a;"da;  =  -^^^^^^^^^ ^ —  CoTx^-^dx. 

J  mloga     mloga*/ 

ra'dx a^ logg    Ca'dx 

J     of  (m  —  Vjof'^     m  —  lJ  a;**"^ 

-.t.,r      r  ai      «    ^       e"*cos"~^a5(acosicH-nsma; 
155.     I  e*''cos"a;daj  = ^^ -— ' . 

or  -\-n^  J 

/rgfn     1 
x^  COS  ax  dx  =   -—  (aa;  sin  aar  +  m  cos  aa;) 


250 


APPENDIX. 


Note  A. 

Assume  that  the  surface  is  given  as  in  Fig.  39,  and  let  the  point  P 
in  Fig.  45  correspond  with  the  point  P  in  Fig.  39. 

PX\  PY',  and  PZ'  are  drawn  parallel  to  the  coordinate  axes.    Let 

PS  be  the  section  of  the  surface  made  by  the  X'Z'  plane,  and  let  PB 

•^  be   the    section    made 

P  m  A      a;'    by    the     Y'Z'    plane. 

Let  PA'  be  the  inter- 
section of  the  plane 
tangent  to  the  surface 
at  P  and  the  X'Z' 
plane,  and  let  PJSf'  be 
the  line  cut  from  the 
same  tangent  plane  by 
the  Y'Z'  plane. 

Evidently  PA'  and 
PJV'  determine  the  tan- 
gent plane  at  P.  Now  let  a  plane  be  passed  parallel  to  the  X'Z'  plane 
at  a  distance  d  below  it.  This  plane  cuts  the  lines  PA'  and  PN'  at  A' 
and  N'  respectively.  A' A  and  N'N  are  drawn  perpendicular  to  PX' 
and  PY'  respectively,  and  the  points  A  and  M  are  connected.  PC  is 
drawn  perpendicular  to  AN,  then  CC  is  drawn  perpendicular  to  A'N', 
and  finally  the  points  P  and  C  are  joined. 

From  the  equation  of  the  surface  by  Art.  42, 

dz_ 
dx 

By 
Let  PA  =  m,  and  PN  —  n. 


and 


tan^P^'  = 


tan  NPN' 


(1) 
(2) 


Now  PC  :m  =  n:  NA  =  n  ;  Vm^  + 


PC 


-y/m'^  +  n^ 


Also 


tan(7PC"  =  ^= 


d       d  Vm^  4-  ^^ 


mn 


hence 


tan2CPC"=— ,H--, 


PO 

tan^^P^'  +  tan^iVPiV^'. 


But 

Therefore 


or 


CPC'  =  y. 


by  (1)  and  (2) 


g)' 


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Jamieson,  C.E.,  F.R.S.E.  Seventh  edition.  Thoroughly  revised  by  W.  J.  Millar. 
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A  Mechanical  Text-Book.     By  Prof.  Macquorn  Rankine  and  E.  F.  Bamber,  C.E. 

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EANKINE,  W.  J.  MACQUORN,  C.E.,  LL.D.,  F.R.S.  Applied  Mechanics.  Comprising 
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REINHART,  CHAS.  W.  Lettering  for  Draftsmen,  Engineers,  and  Students.  A  Prac- 
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RICE,  Prof;  J.  M.,  and  JOHNSON.  Prof.  W.  W.  On  a  New  Method  of  obtaining  the 
Differential  of  Functions,  with  especial  reference  to  the  Newtonian  Conception  of 
Rates  of  Velocities.     12mo,  paper .50 

RIPPER,  WILLIAM.  A  Course  of  Instruction  in  Machine  Drawing  and  Design 
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SHIELDS,  J.  E.  Notes  on  Engineering  Construction.  Embracing  Discussions  of  the 
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SNELL,  ALBION  T.  Electric  Motive  Power:  The  Transmission  and  Distribution 
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Applications  of  Electricity  to  Mining  Work.    8vo,  cloth.    Illustrated.     .     .     $4.00 

STAHL,  A.  W.,  and  WOODS,  A.  T.  Elementary  Mechanism.  A  Text-Book  for 
Students  of  Mechanical  Engineering.  Fourth  edition.  Enlarged.  12mo,  cloth,  $  2.00 

STALEY,  CADY,  and  PIERSON,  GEO.  S.  The  Separate  System  of  Sewerage:  its 
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URQUHART,  J.  W.  Dynamo  Construction.  A  practical  handbook  for  the  use  of 
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Field  Magnet  and  Armature  Winding  and  Grouping,  Compounding,  etc.,  with 
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WILSON,  GEO  Inorganic  Chemistry,  with  New  Notation.  Revised  and  enlarged 
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WRIGHT,  Prof  T.  W.  Elements  of  Mechanics,  including  Kinematics,  Kinetics,  and 
Statics.    With  application.    8vo,  cloth $  2.60 


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